LIBRARV EGRESS. 

• MERICA. 



INDUSTRIAL SCIENCE DRAWING. 

ELEMENTS OF PLANE AND SOLID 
FREE-HAND GEOMETRICAL 

DEAWfflG, 

"WITH 

LETTERING; 

AND SOME ELEMENTS OF 

GEOMETRICAL ORNAMENTAL DESIGN, 

INCLUDING THE PRINCIPLES OF HARMONIC ANGULAR RATIOS, ETC. 



IN THREE PARTS: 
Part I. — Plane Drawing, or from " the flat." 
Part II. — Solid Drawing, or from "the round. 
Part III. — Elements of Geometric Beauty. 



FOR DRAUGHTSMEN AND ARTISANS; AND TEACHERS AND STUDENTS OF 
INDUSTRIAL AND MECHANICAL DRAWING. 





q 1 .A 



Sr EDWARD WARREN, C.E., 

former professor in the rensselaer polytechnic institute, and mass. inst. of 

technology; and boston normal art school; and author of a complkte 

series of text-books on descriptive geometry and instrumental drawij 



NEW YORK: 

JOHN WILEY k SONS, 

15 Astor Place, 

1878 







COPYRIGHT, 

WILEY & SONS, 
1878. 




Trow's 

Printing and Bookbinding Co., 

205-213 East 12th St., 

NEW YORK. 



CONTENTS 



PAGB 

Preface vii 

Preface to the Second Edition xi 



PART I. 

PLANE DRAWING. 

CHAPTER I. 

Exercises on directions of straight lines 1 

First principles 1 

Materials 2 

Directions 2 

Single lines 3 

Parallels 5 

Opposite lines , 5 

Donble lines 6 

Use of the copy-book. Size of figures 6 

CHAPTER II. 

Elementary and practical exercises on right angles 7 

Principles 7 

Examples of single lines at right angles, with sides horizontal and vertical 8 

With sides oblique 9 



IV CONTENTS. 

PAGB 

Right angles 9 

Pairs of parallels at right angles. The pairs horizontal and vertical 10 

The pairs in oblique positions 10 

Practical examples 10 

Occasions for free-hand sketching 11 

CHAPTER III. 

Distances, and division of straight lines 12 

Principles 12 

Exercises in marking off a given distance 12 

Division of lines into equal parts 14 

Practical applications 15 

Enlargement and reduction 18 

CHAPTER IV. 

Circles and their division 20 

Principles 20 

Examples. Circles and Arcs 20 

Division of circles 22 

CHAPTER V. 

Proportional angles 23 

Principles 23 

Elementary examples 24 

Practical examples 26 

CHAPTER VI. 

Figures bounded by straight lines 27 

Principles 27 

Elementary examples 28 

Practical examples 31 

CHAPTER VII. 

Rectilinear and circular combinations 33 

Principles 33 

Exercises 34 



CONTENTS. 



CHAPTER VIII. 

PA OR 

Curves, and curved objects in general. Ex. (103-118) 37 

CHAPTER IX. 

Lettering- , 47 

General principles 47 

Roman capitals 48 

Letters in general. Their classification and construction 51 

Practical remarks 54 



PART II. 

SOLID DRAWING. 

CHAPTER I. 

Object, or model drawing. Rectilinear models 57 

Definitions and principles 57 

Exercises 59 

Curvilinear models 60 

CHAPTER II. 

Perspective and projection free-band drawing 62 

Definitions 62 

Indicated exercises. Properties and treatment of wood 63 

CHAPTER III. 

Pictorial projection sketching 66 

Definitions and principles 66 

Isometrical drawings 66 

Practical applications 68 



VI CONTENTS. 

PART III. 

ELEMENTS OF GEOMETRIC BEAUTY. 
CHAPTER I. 

PAGE 

Elementary ideas. Unity, Variety, Freedom 73 

CHAPTER II. 
Numerical and geometrical expression of the elementary ideas 78 

CHAPTER III. 

General applications of the idea of beauty in ratios. Analogy of form 

and sound 84 

CHAPTER IV. 

Application to triangles and rectangles 90 

Triangles 90 

Rectangles 93 

CHAPTER V. 
Geometric beauty of polygons. Geometrical design 96 

CHAPTER VI. 
Curvilinear geometric beauty. Circles and ellipses 102 

CHAPTER VII. 

Curvilinear geometric beauty. Ovals 110 

Natural and artificial curves 110 

The egg-form, or oval Ill 

Industrial applications 120 

The method by co-ordinates 123 

CHAPTER VIII. 

Geometric symbolism 126 

Definitions and" general illustrations 126 

Geometric illustrations 127 



FROM THE ORIGINAL PREFACE. 



In geometrical, or mechanical drawing, exclusive reliance 
for accuracy may, in theory, be placed on good drawing 
instruments, properly used. 

Practically, these instruments are not absolutely perfect as 
means to the ends to be accomplished by them, and from this, 
and momentary negligences of the draftsman, they are not in- 
fallible in accuracy of operation. 

But, viewing the eye and hand simply as instruments for 
executing conceptions of form, they are incomparably more 
perfect and varied in their capacities in this respect than draft- 
ing instruments ; and well directed practice should, and will, 
bring out this capacity. 

Hence, other things being the same, he will be the most 
expert and elegant draftsman, whose eye is most reliable in its 
estimates of form and size, and whose free hand is most skilled 
in expressing these elements of figure. 

Accordingly, in harmony with the law of easy gradations and 
connecting links which pervades nature, we find a special branch 
of free drawing which is peculiarly well adapted for a prelimi- 
nary training of the eye and hand of the geometrical draftsman. 
This training consists simply in drawing various single and 
combined geometrical lines and figures, of various forms and 
sizes, by the unassisted hand ; and constitutes a connecting link 
between ornamental free drawing and instrumental drawing. 

These brief reflections have resulted from a recent inspection 



VI 11 PREFACE. 

of a few simple pencil plates of such drawings forgotten for a 
long time, having been made by the writer several years since, 
in connection with the conduct of a short course of exercises of 
the kind above described. 

As a further, and I hope not useless fruit of the foregoing 
views, the following little course is presented to all who, as 
draftsmen, may promise themselves benefit from the use of it, 
and for exercises of mingled interrogation and practice in pre- 
paratory and industrial schools. 

Ey means of a love of skill and accuracy in the use of eye 
and hand, exercises like those of this volume may be made 
a pastime for the improving (especially if social) enlivenment 
of numerous odd moments, those times when many subordinate 
excellencies can be acquired or perpetuated without interference 
with one's larger industries. 

Writing, as merely auxiliary to daily business, is not, in its 
intention, a branch of drawing. But, as an ornamental art, it 
is a species of free-hand drawing, not geometrical, however. 
Hence I have not treated of writing, while ample instructions 
on lettering have been deemed a due portion of the contents of 
this volume, since, moreover, the usual small size of letters 
makes their construction by hand alone more convenient than 
by the use of drafting instruments. 

The good tendencies of accurate drawing in regard to mind 
and character are worthy of notice. Practice in such drawing 
directly tends to make close and accurate observers, who will 
thus gain distinct conceptions of the objects of attention, and 
so of thought generally, and who will then more readily pass on 
to fidelity in the representation of their observations and con- 
ceptions. 

Newton, Mass., January, 1873. 



NEW PREFACE. 



In distinguishing between artists proper ; and those engaged 
in the study or practice of industrial design — that of various 
wares and fabrics — together with those engaged in engineering 
and mechanical study or practice, including instrumental 
drawing ; it seems appropriate that both the latter classes 
should receive a special training — useful also to all — in the 
free-hand drawing of regular forms. Hence my former 
small work was put forth, partly as an experiment. Increasing 
interest in the subject, and the measure of favor accorded to 
that less complete volume, has induced the author to studiously 
revise, remodel, and enlarge his work, adding many new figures, 
mostly on plates. 

Of the three Parts, into which the present volume is divi- 
ded, Part II. is largely, and Part III., almost wholly new. 

Part III. may interest the general reader. It contains, ap- 
parently the most appropriate principal extension of the volume, 
a concise presentation of the elements of geometric beauty, 
based, in general, on the ingenious and presumably correct 
theory of D. R. Hay ; but containing principles and applica- 
tions not found in his " Principles of Symmetric Beauty." 
Especially, the ovals, or egg-forms, derived naturally, and in 
unlimited variety, are believed to be new, and an improvement 
upon his " composite ellipse." 

The subject of symbolism may, in some aspects, of course, be 
turned into pleasantry. Still, as its use has prevailed for cen- 



X PREFACE. 

turies in some departments, there seems to be no reason for not 
extending it to others. I have sought to simplify, and to 
improve as much as possible, the very little that room could be 
found for, on this subject. 

The figures or patterns in this volume may have a threefold 
use : Firsts merely as copies for imitation, in acquiring skill of 
eye and hand. Second, as standards, from which to make as 
many variations by recombination of elements, as ingenuity 
can invent. Third, as objects to which to apply the principles 
of beauty developed in Part III. Bearing this in mind, it will 
be seen, that the appropriate range of use of this volume may 
extend through Public and Private Preparatory Schools, Arti- 
zan's and other Evening Schools ; Schools of Design, and the 
earlier classes in Polytechnic Schools. 

The pupil's figures may conveniently be drawn in blank copy- 
books, easily procurable, and, in most cases should be consider- 
ably larger than the copies, in order to cultivate a broader 
freedom of movement of the hand. 

By a new process, enabling the plates to be close imitations 
of the autograph originals, the rigid straitness of ruled lines, 
which could not well be imitated by the free-hand, is avoided ; 
and the copies are such as the pains-taking pupil may reason- 
ably be expected to equal, and encouraged to excel. 

Grateful for the favor long accorded to his other elementary 
works, the author, ever bearing in mind, and recommending 
joint attention to principles and practice, hopes to make them 
still more acceptable by extending the work of thorough re- 
vision (for the first time, excepting the Projection Drawing) to 
all of them. 

Newton, Mass., August, 1878. 



FREE-HAND GEOMETRICAL DRAWING, 



PART I. 

PLANE DRAWING. 



CHAPTER I. 

EXERCISES ON DIRECTIONS OF STRAIGHT LINES. 

First Principles. 

The direction of a straight line is its invariable tendency to- 
wards some fixed point. 

The directions of two lines may be alike. The lines are then 
said to have the same direction, and are called parallel. 

The drawing of parallel lines, or those whose directions are 
alike, is simpler than that of lines whose direc- 
tions are different, and hence is here considered 
first. 

A line which is " straight np and down," or 
perpendicular to the surface of water, like this, 
when the book is held upright, is called vertical. 
A level line is called horizontal. 

The force of gravity acts vertically, hence objects rest with 
most stability in a vertical position ©n horizontal surfaces. 
Likewise, man himself, naturally stands upright, or vertically, 
and, generally, on surfaces whose lines are level, or horizontal. 
Hence vertical and horizontal are the simplest, most familiar, 
or primitive directions of lines, and will be first considered. 

Lines which are neither vertical nor horizontal, are oblique. 
Also lines lying in any fiat surface, and not representing either 
of these positions, are called oblique. 

Before commencing the succeeding exercises, the learner 
should be provided with the following materials ; and, through- 
out his progress, should carefully follow the subjoined general 
directions. 



2 FREE-HAND GEOMETRICAL DRAWING. 

Materials. 

For the practice of quite young pupils, where substantial ac 
curacy, rather than fineness of execution is expected, quite 
cheap paper, or even a slate and pencil will answer. 

For other pupils, blank drawing books of the usual form, 
everywhere easily obtained, may be used ; or, drawing paper 
may be cut into plates of convenient size, and kept in a paper- 
case such as any one can make for himself. 

A common semi-circular "protractor" a semi-circular piece 
of thin material, divided into degrees on its curved edge. 

A ruler 10 inches long and 1 inch wide. 

Moderately soft pencils, as Faber's No. 2 and 3. 

Prepared india rubber, free from grit, of the best kind now 
known as " Artist's gum." 

Spare pieces of paper, one, on which to rest the hand and so 
protect the drawing, and another on which to try the pencils. 
Also a strip for a measure of distances. 

A fine file, on which the pencil can, by a rolling rubbing 
motion, be most neatly and readily sharpened to a round point. 

When accuracy on a large scale is sought, as a training for 
bold sketching, plates, 10 ins. by 14 ins. of buff manilla drafting 
paper, and crayons, should be substituted for pencils, and small 
plates and figures. In fact, this may be done as a preliminary 
counterpoise to the somewhat cramping tendency of the mostly 
minute accuracy required in mechanical drawing. But for 
direct training in this accuracy, the pencils, and small plates, 
should be used as above indicated. 

Directions. 

Depend on the unassisted eye and hand alone, from the be- 
ginning. They w T ill, in due time, amply reward the reliance 
placed upon them. Ruler, Protractor, and Measure may be 
used to test the straight ness, direction and length of lines al- 
ready drawn, so that if incorrect they can be re-drawn. But 
they should never be used to locate, limit, or rule the lines; for 
thus no education is afforded to the eye and hand, only trifling 
skill is gained by them, and so the main object of the exercises 
is missed. 

If a line is found incorrect, first consider carefully how it 



DIRECTIONS OF STRAIGHT LINES. 3 

differs from what it was meant to be, then erase it, and study its 
direction well, and try again. Excellent quality, and not great 
quantity of drawing, is to be the chief object of ambition. 

Avoid the use of the rubber by studying well the position 
and lengths of the lines before drawing them. Mean to have 
them appear in a certain way, and then make them so, as truly 
as possible ; rather than hastily make a careless sketch and then 
seek how to correct it. 

Be sure that a figure is as well done as possible at the time, 
in obedience to the preceding rules, before attempting a new 
figure. 

Hold the pencil between the thumb and forefinger, and rest- 
ing on the tip of the second finger. It can then be moved both 
with freedom and steadiness. 

In drawing lines towards or from yon, let the elbow be at 
some distance from the body. In drawing lines from side to 
side let the elbow be close to the body. 

Arrange the seat and paper so as to look at the paper in a 
direction at right angles to it, without stooping, and let the desk 
be low enough not to interfere with the elbows. 

Though all the lines of the following figures are horizontal, 
when the book lies flat, yet, for the sake of brevity, it may be 
understood that all those lines shall be called vertical, which are 
so when the book is held vertically. Lines from side to side 
may be called horizontal, and others, oblique. 

Remember especially to sketch each of the figures, first in 
very faint lines, which can easily be erased if incorrect, before 
drawing the firm heavy lines of the finished figures. Do not, 
however creep along the line by short, disconnected, and 

hesitating steps, thus, '* . ; but mark the 

line by a firm and unbroken movement, first lightly, thus, 
and then heavily, thus: 

Single Lines. 

Example, 1. Draw vertical lines, beginning at the top, and 
far enough apart to prevent each from being a guide to the 
other, as a parallel. Thus let these lines be drawn at the 
middle and ends of the upper half of the plate. See A, PI. I. 

Ex. 2. The same on the lower half of the plate, but beginning 
the lines at the bottom. See PL I. 



4 FREE-HAND GEOMETRICAL DRAWING. 

Ex. 3. Mark two points so as to be connected by a vertical 
line, and then draw a line joining them, beginning a little above 
the upper point. 

Ex. 4. The same, but beginning below the lower point. 

These, and all the examples, should be varied, by taking lines 
of various lengths. 

Ex. 5. Draw horizontal lines, beginning at the left, and far 
apart, as at the top, middle and bottom of the left hand half of 
a plate. See PL I. 

Ex. 6. The same on the right hand half of the plate, and 
beginning at the right. This will require special care. 

Ex. 7. Mark two points so as to join them by a horizontal 
line, beginning to the left of the left hand one, and draw to the 
right. 

Ex. 8. The same, only begin to the right of the right hand one. 

The foregoing constructions not all shown on PI. I. will 
divide the plate into quarters, in which the following may be 
drawn. 

Ex. 9 to 12. May consist of the four preceding variations in 
the manner of drawing, applied to an oblique line, which in- 
clines from the body and to the rights thus : 



Ex. 13 to 16. May consist of four similar constructions of 
lines which incline from the body but to the left. 

The last two examples should also be practised with the two 
following variations : First, let the lines be more nearly vertical 
than horizontal, thus : 



DIRECTIONS OF STRAIGHT LINES. 



Second, let them be more nearly horizontal than vertical, 
thus : 



Parallels. 

The following Examples permit so many variations in the 
order of construction, that each one, as numbered, must be 
generally understood to include several particular varieties. 

Ex. 17. Draw two vertical parallels, first drawing the left 
hand one first; and second, the right hand one first. Also 
draw each, in the four ways described in Examples 1 to 4. 

Ex. 18. Likewise draw two horizontal parallels, first, drawing 
the upper one first ; and second, the lower one first, and each as 
in examples 5 to 8. 

Ex. 19. Draw several vertical parallels, beginning alternately 
at top and bottom. See a., PL I. 

Ex. 20 to 22. May consist of similar variations in drawing 
two or more horizontal parallels. See h., PL I. 

Ex. 23 to 28. May consist of similar exercises on two or 
more oblique parallels situated as in examples 9 to 16, and in- 
cluding the variations in the amount of obliquity there pointed 
out. 

Opposite Lines. 

These are lines starting at a given point ; and proceeding in 
opposite directions, thus ; 



+ ■ 



or towards each other from their outer extremities. 

Ex. 29. Draw opposite lines, one upwards, and one down- 
wards from the given point. 

Ex. 30. Do., one to the right, and one to the left of a given 
point. 

Ex. 31. Do., in the principal varieties of oblique position. 

Ex. 32. Is a comprehensive one, consisting of the variation 
of the three preceding, by beginning to draw the opposite lines 
in each case from their outer extremities. 



6 FREE-HAND GEOMETRICAL DRAWING . 

Double Lines. 

All the preceding examples may be made in double lines ; 
that is, lines as close together as they can be made without 
touching, and at first of the same size, and then, of different 
sizes. 

Useful practice under this head consists in filling various 
figures, such as triangles, squares, polygons and circles, with 
parallel lines, which should be made equidistant by the eye. 

General Example. Construct a series of examples of figures 
thus filled, each with one, two, three, or four sets of parallels ; 
which will form an elegant imitation of bold line engraving. 
See PI. I, Fig. 1. 

Use of the Copy-book— Size of Figures. 

The figures, many of which are, for convenience, printed of 
small size, and with the text where they are described, should 
be considerably enlarged as drawn in the copy-book, by the 
pupil. 

The plates give a better idea of the size and style of the 
figures as they should be drawn. Only, as the pupil's plates 
may be more numerous and less crowded, his figures may be 
larger at pleasure, making from one to six to a plate, according 
to their character. 

Indeed, where the figures are done with crayons, they should 
be made much larger, and may each be made to fill a buff 
manilla paper plate twice as large as those of the copy-book. 
Plates IX. and X. contain such figures ; and when so drawn, 
they cultivate a greater freedom of movement of the hand, com- 
bined with exactness, than is secured by the finer work with 
the pencil alone. 

To avoid injurious wear of the copy-book by repeated trials, 
it may often be best to draw the figures first on loose waste 
plates. 



CHAPTER II. 

ELEMENTARY AND PRACTICAL EXERCISES ON EIGHT ANGLES. 

Principles. 

Beauty of form, considered as residing in certain geometrical 
properties of regular figures, results from certain proportions 
between their parts. These proportions may be regarded as 
arising from the relative lengths of the distinguishing lines of 
the objects ; or from the relative sizes of their angles. 

In moving, whether to walk, or to merely draw a line, we 
must begin each movement at a given point. The direction of 
our movement is first in our thoughts, rather than its extent. 
We first, if not oftenest, think, or ask, " which way " than " how 
far." Direction is therefore a more primary idea than length. 
An angle, however, is merely difference between directions 
from a certain point. Hence angular proportions, or the pro- 
portion between the angles of a figure, are more elementary than 
linear proportions, or those between the lengths of the lines of 
the figure, and will be first considered. 

In doing this it will be convenient to find first some angle as a 
natural standard of comparison for all others, and this we now 
proceed to do. When, then, two lines are so situated that, in 
moving on one of them, we do not at all move in the direction 
of the other, their directions are said to be independent. 

V h 




m 



-k 



c f y 

Thus, in these figures, by going from a to b, we find our- 
selves at the distance ac to the right of a. So by moving from 
d to e we go a distance equal to df in the direction of the line 
df But, when the two angles formed by the meeting lines are 
equal, as at mgh and kgh, we do not, in moving to any distance 
on gh, progress at all to the right or left of gh. Hence the 



8 FREE-HAND GEOMETRICAL DRAWING. 

directions of gh and mk are independent, and the angle included 
between them is the naturcd standard with which to compare 
all other angles. This angle between independent directions is 
called a right angle / and now some of the subsequent exercises 
are to consist in constructing, by the eye, various proportional 
parts of a right angle. 

But, again, it follows from the explanation of vertical and 
horizontal directions, in Chapter I., that a right angle is in its 
simplest, most natural, or standard position,wh.en its sides are in 
the fundamental directions of vertical and horizontal. We there- 
fore begin with right angles in this position. Observe first, how- 
ever, that we do not say perpendicular and horizontal, but vertical 
and horizontal, for a line in any position whatever, is perpen- 
dicular to another when it is at right angles with it, but there is 
but one vertical, or " straight up and down " direction. 

Examples of Single Lines at llight Angles, with Sides Hori- 
zontal and Vertical. 

Ex. 33. Construct one right angle thus, | t and thus, 

and thus, and thus, making its sides 

from one to three inches in length, each side ending at its inter- 
section with the other. Slight additions will give these simple 
elementary figures a pleasing character as designs for geometri- 
cal borders and corner pieces, thus : 



L 



designs which it is easy to make evenly by observing the direc- 
tion to pencil each line faintly at first, while locating it as in- 
tended. 

Observing that the beauty of a border depends upon its 
expressiveness, as an echo of some characteristic of the work 
which it encloses, the first design would make an agreeable 



EXERCISES ON RIGHT ANGLES. 



9 



corner, for a plate of figures made up of points and straight 
lines. The second, with its swelled lines, suggests strength in 
the corner of the border; which can also be gained by a dia- 
gonal from the outer corner to, or a little beyond the inner 
corner. 

Ex. 34. Construct two right angles, by prolonging one of the 

and thus, 




> 



sides beyond the vertex of the angle, thus, 
and thus, and thus, — I 

making the sides of each angle from 
two to four inches long, in this and 
the following figures. 

Ex. 35. Construct four right angles, by prolonging each side 
through its point of intersection with the other, thus 

Right Angles with sides Oblique. 

Ex. 36. Repeat Ex. 33 with the sides in various oblique posi- 
tions, and of various lengths, thus : 

Ex. 37. In like manner, repeat Ex. 34, thus : 

-l -r y y- 

Ex. 38. Similarly, repeat Ex. 35, thus, but in each case make 

the lines from one to three or four 
inches long, from the point of inter- 
section. 

See c and d, PL I., for examples 
of a suitable size for these figures, 
but which, if drawn on large plates, 
may be larger still. 




10 



FREE-HAND GEOMETRICAL DRAWING. 



PAIRS OF PARALLELS AT RIGHT ANGLES. 



The Pairs Horizontal and Vertical. 

Many variations can be made, and shonld be, in the order of 
drawing each, of the following figures. Thus the vertical lines 
can both be begun at top or bottom, or one in each way ; alsc, 
the horizontal lines may both be begun at the left end, or right 
end, or one in each way. Again, both of the vertical lines 
may be drawn first, or both of the horizontal ones, or one of 
each in succession. 

Ex. 39. To give a more ornamental character to these simple 
elements, after seeking truth pi representation, only, in the pre- 
ceding elementary figures, they may consist in combinations of 
faint and heavy lines, as shown in a part of the following fig- 
ures, all of which should be made of Lines from a half inch to 
three or more inches long. 



□ 



L 



L 



+ 



n 



The pairs in oblique positions. 
Ex. 40. Eepeat Ex. 39, as follows : 





Practical Examples. 

Ex. 41. The preceding elementary examples afford all the 
operations necessary in forming many simple drawings, either 
of geometrical designs for surface ornament^ or of objects. 



EXERCISES ON RIGHT ANGLES. 

A specimen or two of each is added in this example. 



11 



| >ijgfP 




_4 


|,*".< 


jgi, IN*! 


il 




The pupil is here again reminded always to make his figures 
very much larger than those of the book, making from two to 
four on a plate like those here shown. 

Ex. 42. Figs. 2, 3, 4, PL I., exhibit other practical examples 
of constructions containing only lines at right angles to each 
other. 

Occasions for Free-hand SJcetching. 

The travelling student, architect, engineer, mason or builder 
may often find it desirable to make hasty sketches of neatly 
contrived details or structures, whether in masonry, wood, or 
metal. So also may persons of any and all occupations, when 
seeking to " give an idea " of something which they wish to 
have constructed ; and, in both cases, drafting instruments and 
time to use them may not be at command. 

The examples mentioned, and many similar ones which they 
may suggest, or which may usefully be collected by observation, 
should therefore be carefully drawn on various scales by the 
pupil, as a means of acquiring skill in the useful accomplish- 
ment of readily making free-hand sketches of geometrical in- 
dustrial objects. 



CHAPTEK III. 

DISTANCES, AND DIVISION OF STRAIGHT LINES. 

Principles. 

Having considered various directions of straight lines, we are 
prepared to estimate and represent various distances npon them. 

Distances are eqtial or unequal. When unequal, we often 
wish to compare them. Distances may be compared, first, by 
taking one from the other, and thus finding their difference. 
This shows how much greater, or smaller, one distance is than 
the other. 

Distances may also be compared, second, by observing how 
many times one is contained in the other, and thus finding their 
ratio. This shows how many times greater the larger distance 
is than the smaller, or what part the smaller is of the greater. 

When we compare lines in this second way, we speak of them 
as proportional, or as being in proportion to each other, or as 
having a certain proportion to each other. 

An indefinite line is one that has no given limits. In repre- 
senting distances, we may either mark a given distance several 
successive times on an indefinite line; or, we may divide a 
given line into equal parts, and so find a series of equal distances. 

Exercises in Marking off a Given Distance. 

Ex. 43. Draw straight lines in different directions, and mark 
by the eye, the same distance, once, on all of them, thus : 



DISTANCES AND DIVISION OF STRAIGHT LINES. 



13 



Transfer the distance on the first line to the edge of a slip of 
paper, and with this, as a measure, see if the distances on the 
other lines all agree with this measure. If not, observe whether 
they are too large or too small, and then, without making any 
mark on the paper before removing the measure, take away the 
measure, and correct the distances by the eye. 

Ex. 44. In like manner, mark a given distance several times, 
on lines in various directions ; thus : 




Ex. 45. Draw lines in several directions through the same 
point, and mark equal distances from the point on all of them ; 
thus: 




14 FREE-HAND GEOMETRICAL DRAWING. 

Division of Lines into Equal Parts. 

Ex. 46. Divide lines in various positions, as shown below, intc 
two equal parts. This is done by marking the middle point of 
the line, and is called bisecting the line. Then apply the paper 
measure, and see if the two parts are equal. If they are not, 
the error found at the end of the line will be double the error 
in the required half. If three parts had been required, this 
final error would have been three times the error in the single 
third of the line, and so on. Then make the necessary correc- 
tions, accordingly. 



To distinguish these figures from the preceding, mark only 
the ends of the line by dashes extending across the line. 

Ex. 47. Divide a line into four equal parts. To do this, 
bisect the whole line, and then bisect each of its halves. 



In each of these exercises, let the given line be taken in 
various positions, though but one may be shown in the book. 

In like manner, that is, by bisecting each quarter of a line, 
we should obtain eight equal parts, etc. 

Ex. 48. To divide a line into three equal parts, that is, to trisect 
it. Estimate one-third of the line, and bisect the remainder. 



To divide a line into nine equal parts, divide each of its 
thirds into three equal parts. 

Ex. 49. In the preceding examples, we have divided each of 
the larger spaces into the same number of parts into which the 
whole was first divided. 



DISTANCES AND DIVISION OF STRAIGHT LINES. 15 

Let a line now be divided into six equal parts, for example. 
Half of a line is more easily estimated than a third, hence di- 
vide the line first into halves. Also, having done this, one-third 
of a short distance, as the half line, is more easily estimated 
than a third of the longer whole line, hence divide each half 
into thirds, giving six equal parts in the whole line. 

Ex. 50. To divide a line into any prime number, as five, 
seven, eleven, etc., of equal parts, it is necessary to estimate at 
once the fifth, seventh, eleventh, etc., part of the whole line. 
Yet this may be done more readily by dividing the line into 
two or more parts. Thus, one third of a line to be divided into 
seven equal parts would contain two and one-third of those 
parts, and thus we could more easily estimate the size of one 
of those parts. 

Practical Applications. 

PL II. gives examples of various useful exercises in distance, 
direction, division. 

In order to enter upon the drawing of these figures, and 
many similar ones, with proper ideas and spirit, it is neces- 
sary to understand that, although they would, in final and 
finished practice, be drawn with instruments, yet it is highly 
useful to draw them also first by the eye, and for various 
reasons, such as follow. 

First ; it may (see p. 11) often be desirable to make rapid 
sketches, when instruments are not at hand. Second ; instru- 
ments are liable to be displaced unconsciously to the draftsmen, 
giving unequal spaces, if dividers are jarred, or a scale mis- 
takenly used; and untrue directions, if a ruler be displaced. In 
these cases, the eye may be so accurately trained as to readily 
detect errors, which if long undiscovered would, and do, occa- 
sion great annoyance. Third ; the eye so trained will often 
enable the draftsman to make small and simple divisions, espe- 
cially those requiring only repeated bisection, as halves, fourths, 
etc., as readily and perfectly by the eye, as with the com- 
passes. 

All the following figures were originally drawn strictly as 
here directed, the divisions being tested by marking them on 
the edge of a slip of paper, and the directions, see especially 
the diagonals in Fig. 6, by tracing them at first very faintly. 



16 FKEE-HAND GEOMETRICAL DRAWING. 

The pupil should therefore begin with an effectual ambition 
and purpose to perfect his work without instruments. 

Ex. 51. Customary black-hoard exercises. — Besides figures 
drawn on the black-board only for varied practice, or some 
other special purpose, various common subjects of study cus- 
tomarily require the drawing of numerous black-board diagrams. 
It should never be supposed, that because these diagrams are 
temporary, they may be carelessly drawn. On the contrary, on 
the teacher's part, neat diagrams lend interest to explanations, 
and naturally stimulate the learner to draw equally good ones ; 
while on the pupil's part, the pains taken in making them helps 
to form the invaluable habit of doing as well as possible what- 
ever one does. 

Accordingly, Fig. 1 is a diagram, connected with the use of 
instruments, and Fig. 2, is from plane geometry. 

Fig. 1 represents what is called a diagonal scale, from its 
diagonal lines at 1, 2, 3. A scale is a contrivance for repre- 
senting any actual measure, as a foot, yard, or mile, by some 
other measure, usually a smaller one. Thus, let the distance 
from 0, to 1yd., be two inches, but let it represent one -yard. 
It will then be called one yard. The next lower denomination 
is feet, hence, making the distance 0, 3ft. equal to two inches 
divide it into three equal parts, each of which will therefore 
represent a foot, and will be called a foot. Now suppose we 
wish to represent fourths of a foot, or three-inch spaces. Draw 
five equi-distant parallel lines, at any convenient distance apart, 
as shown, and divide the distance 0', 3' on the lowest line, into 
three equal parts, and draw the diagonals as shown. Then you 
see that as 0'a is one fourth of O'O ; ac is one fourth of 01. 
But 01 is one foot, hence ac is one fourth of one foot or three 
inches. 

In like manner, the distance between the two heavy dots is 
1 yard, 1 ft., and nr, which is three-fourths of 01, or 9 inches. 

As a drawing-exercise, the points to be observed are to make 
those lines straight and parallel, and those divisions equal, that 
are intended to be so. 

The pupil can exercise his ingenuity in making other diago- 
nal scales from the full description given of this one ; as, for 
example, one of feet, inches, and half-inches ; or one of units, 
lOths, and lOOths. 



DISTANCES AND DIVISION OF STRAIGHT LINES. 17 

In Fig. 2, the exercise consists in making ab perfectly paral- 
lel to AB, and cd to AD ; and in drawing Ab straight from A 
to b and A^, from A to d. Then, triangles, like AaB, and A3B, 
having the same base, AB, and equal altitudes (the perpendicu- 
lar distance betweefl ab and AB) have equal areas. Also, as 
the like is true of the triangles AcD, and AdD, the triangle 
Abd is equivalent to the polygon AaBDc. 

Ex. 52. Floor and wall decoration. — PL II., Figs. 3, 4, and 
5, are examples requiring equal divisions of one or more sizes, 
and parallel lines in different directions. 

Fig. 3, represents diagonal floor- work in two w T oods, the con- 
struction being founded upon the square 0, 6, 6, S. Divide 
the sides of this square into any desired even number of equal 
parts, and draw Sd and the parallels to it through the corners 
and middle points of the sides of the square to form the paral- 
lel bands of flooring ; each band being filled with narrow diag- 
onal strips. These strips are drawn parallel to the sides of the 
square, through points of division on an adjacent side ; as jpy, 
parallel to 6, 6, through 5 on 0, 6. 

Finally, the two woods may be arranged in two ways ; first, 
touching each other on a common edge, as at ab ; second, 
touching only at the corners, as at c and d. 

Fig. 4, represents a toothed cornice, the shaded portions giv- 
ing the effect of shadows. 

Fig. 5, represents an ornamental band of triangular points, 
the darker portions indicating a darker color. In dividing, 
equally, the top and bottom lines, be careful to make the points, 
as a, of the triangles exactly over the middle points, as b, of 
their bases. 

Ex. 53. Fig. 6, gives further occasion for practice in com- 
bined equal division / in the equal bars, and in the larger 
equal spaces between them. Also in the direction of the diag- 
onal brace. 

In this, and all similar cases, the divisions, first made only 
by the eye (p. 2, Directions), may be adjusted by the aid of a 
slip of paper to the edge of which a division of each kind can 
be transferred and used to test the others. 

The directions are adjusted by sketching them very faintly, 
just skimming the point of the pencil along the paper, until 
they are found to be correct, when the faint traces thus ob- 



18 FREE-HAND GEOMETRICAL DRAWING. 

tained can be firmly drawn in heavier lines with a softer 
pencil. 

Enlargement and Reduction. 

PL II., Figs. 7 and 8, and Fig. 9, exhibit two methods of ac- 
complishing an often desirable and useful purpose, that of 
reducing a given figure in any desired ratio. One of these 
may be called the method by subdivision, the other, that by 
concentration. 

Ex. 54. Fig. 7, may represent a frame in the form of a 
capital "A," carrying a plummet, and standing in a window 
opening of irregular outline. Fig. 8 is a reduced copy in 
which each side of the auxiliary enclosing square is five- 
eighths of that in Fig. 7. By subdivision of this square into 
the same number of smaller squares in each figure, the por- 
tion of the figure embraced in each small square will be so 
small that it can be very accurately drawn by the eye. 

This method may be applied to the sketching of large ob- 
jects, by substituting for the subdivided square of Fig. 7, a 
frame of equidistant threads, crossing each other so as to form 
squares. Then by setting this frame in some suitable fixed 
position, and viewing the given object through it, from some 
fixed point, the portion of the object seen in each thread 
square can be traced in the corresponding compartment of the 
similarly subdivided square on the paper. 

When, however, the ability already exists to draw objects 
accurately from the original, this method by the thread frame 
is unnecessary. And where the purpose is to acquire this 
ability by a sufficiently extended practice in sketching from 
original objects, the same method might only hinder the re- 
sult. But in the many cases still remaining, the method will 
be found useful.* 

Ex. 55. Fig. 9, represents a given irregular figure, the outer 
one, reduced to the smaller one by means of the auxiliary lines 
which converge from its angles to 0. The truth of the method 
will be apparent by conceiving to be the vertex of a pyramid 
whose base is the given figure, and the reduced figure to be a 
section of this pyramid, parallel to its base. Each side of the 

* See my Elementary Perspective, Part II., Chap. IV. 



DISTANCES AND DIVISION OF STKAIGIIT LINES. 19 

smaller figure will then be parallel to the corresponding side 
of the larger one, and all of the converging lines will be 
divided in the same manner. That is, if any one of them be 
bisected, as in the figure, all of them will be bisected. Thus 
the method, when executed by the free hand throughout, 
affords practice in three things, first, the drawing of straight 
lines in various directions, each joining two given points; 
second, in drawing parallel lines ; third, in the equal division 
of lines. 

Pupils can profitably practice extensively on these two 
methods of copying, with variation of size. The former con- 
veniently applies to the copying of geographical maps, carpet, 
paper, and inlaid patterns of regular form, and to letters. The 
latter method applies better to polygons, regular or irregular, 
as the boundaries of the map of a field. 



CHAPTER IV. 

CIRCLES AND THEIR DIVISION. 

Principles. 

Direction is, as before said, tendency towards a certain point 
A straight line has but one direction at all of its points. 

A curve constantly changes its direction. 

The simplest curve, and the one which will be the natural 
standard of comparison for all other curves, is the one which 
changes its direction at a uniform rate. The circle is such a 
curve, and all its points are at equal distances from one point 
within called its centre. The circle is, therefore, the simplest 
curve, and standard of comparison for other curves. 

Examples. Circles and Arcs. 
Ex. 56. To draw a circle. Sketch, faintly, several lines through 
a point, taken as the centre of the circle, and, from this point, 
mark off equal distances on each of these lines. Then through 
the points thus given draw the circle, thus : 




Ex. 57. To draw the circle without drawing the lines through 
its centre. With the paper measure, mark a number of points 
all at the same distance from the centre, and then sketch the 
circle through those points. 




CIRCLES AND THEIR DIVISIONS. 



21 



In both of these constructions, use fewer and fewer guides, 
and at last sketch a circle with no guiding point but its centre. 
Also practice often in rapidly drawing circles by hand on the 
black board. 

The distance from the centre to the circumference of a circle, 
is called its radius. The distance across the circle, through its 
centre is its diameter. 

Parallel circles have the same centre, and are called con- 
centric. A portion of the circumference of a circle, is called 
an arc. 

Ex. 58. Draw circular arcs in various positions, and of various 
radii, and length, thus : 



Ex. 59. Draw parallel arcs and circles, of various radii, and 
the former also of various lengths and in various positions, 
thus ; and then mark their centres. 




22 FREE-HAND GEOMETRICAL DRAWING. 

Division of Circles. 

Circles, or arcs, may, like straight lines, have given distances 
marked off upon them, and may be divided into equal parts. 

The line which joins the extremities of an arc, is called the 
chord of that arc. When the arc is very short, its length cannot 
be ordinarily distinguished from that of its chord. It is on 
this principle that any given straight distance may be trans- 
ferred to a circle or to any curve. 

Ex. 60. To lay off a given distance on a circle or arc, divide 
that distance into a sufficient number of small equal parts, and 
then mark off on the circle, or arc, the same number of similar 
equal parts, thus, where the straight line is the given distance. 



Ex. 61. Any diameter of a circle divides it into two equal 
parts, therefore draw several circles, and one diameter in each ; 
but in different positions in the different circles, which may also 
be of various sizes. 

Ex. 62. Two diameters at right angles to each other, divide a 
circle into four equal parts. Draw such diameters in various 
positions. 

Ex. 63. The radius of a circle applies just six times to its cir- 
cumference. Then lay off the radius once, as a chord of the 
circumference, as explained above, and then mark the other 
divisions, equal to the one thus obtained. 

Ex. 64. Bisect each quarter circle in Ex. 62, which will give 
eight equal parts in the whole circle. 

This bisection can then be continued to any extent, giving 
sixteenths, etc., of the circumference. 

Ex. 65. Continue these exercises by trisecting the quarter 
circles, and bisecting and trisecting the sixth parts in Ex. 63, 
giving twelfths, eighteenths, etc., of the whole circle. Also 
make these divisions on circles of various sizes, and on arcs in 
various positions. The eye will thus be trained to estimate 
readily any given part of a circumference. 



OHAPTEK Y. 

PROPORTIONAL ANGLES. 

Princvpfos. 

After acquiring power to draw lines, truly straight, in any 
direction, and to draw a true right angle in any position, much 
additional power of the eye to estimate, and of the hand to rep- 
resent, will result from practice in estimating the values of the 
angles of objects. But we have seen that the right angle, upon 
which varied practice has now been had, is the natural standard 
of comparison for other angles. Hence the new group of valu- 
able exercises which follow, is designed to train the learner in 
estimating and representing accurately any fractional part of a 
right angle in any position. 

Every circle is considered as being divided into three hun- 
dred and sixty equal parts, called degrees and marked thus, 
360°. Hence a half circle embraces 180° ; a quarter circle, 90° ; 
a sixth of a circle, 60°, etc. But, as already seen in the last 
chapter, two diameters at right angles to each other divide a 
circle into quarters ; hence, as a right angle includes a quarter 
circle, or arc of 90°, between its sides, it is also called an angle 
of 90°. 

In like manner, any angle is said to be an angle of as many 
degrees as there are in the arc between its sides, the centre of 
the arc being at the point or vertex of the angle. In other 
words an angle is said to be measured by the arc included be- 
tween its sides. Hence the easiest way to divide an angle into 
equal parts, or parts having any given proportion to each other, 
is, to divide the arc between its sides in the manner required, 
and then to draw straight lines from these points of division to 
the vertex of the angle. The right angle being, as before ex- 
plained, the natural angular measure for other angles, a right 
angle will be taken as the one to be variously divided, in the 
following examples. 



24 



FREE-HAND GEOMETRICAL DRAWING. 



Elementary Examples. - 

Ex. 66. Bisect a right angle, in each of the positions given in 
Ex. 33. To do this, sketch carefully a quarter circle between 
the sides of the angle and mark the middle point of this arc, 
Then join this middle point with the vertex of the angle as 
seen in the figure. To divide the angle into any other number 
of parts, divide the included quarter circle into the same number 
of parts. To test the angle thus estimated and drawn, use a 



" Protractor," as follows 




The protractor is a semi- circular instrument, whose semi-circu 
lar edge is divided into 180 degrees. A right angle is an angle 
of 90°. Half a right angle is 45°, hence if we place the straight 
side of the protractor on one side of the angle, and its centre, C, 
marked by a notch, at the point or verier, C, of the right angle, 
as shown in the figure, then the required bisecting line C, 45°, 
will if correct pass through the 45° point on the divided edge 
of the protractor. If it fails to do so, then first carefully esti- 
mate, by the eye, the amount of error, and then erase the line and 
draw it over, remembering to sketch it lightly ', till found correct. 
Having found the true direction of the required dividing line 
of the given angle, draw a number of parallels to it, in this, 
and all the following problems of divisions of angles. 

Ex. 67. Construct a line which will cut off one-third of a 
right angle from either of its sides, thus : 





TROrORTIONAL ANGLER. 



25 



( hie-thiid of a right angle is 30° — measured by one-third of the 
quarter circle — hence in testing the lines after drawing them 
they should pass through the 30° point of the protractor in the 
first figure, and the (30° point in the second. In every case con- 
sider, as above, the number of degrees in the given fractional 
part of the right angle, and make the test accordingly. 

In the figure, only two parallels to the required direction are 
shown. The student should make many more, and in various 
positions around the original figure. 

Ex. 68. Draw a line cutting off one-fourth of a right angle 
from either of its sides. This can be most accurately done by 
bisecting half a right angle, thus : 





Observe, as indicated in these figures, to place the given right 
angle in any and all of the positions given in Ex. 33. 

Ex. 69. Cc nstruct, successively, angles of one-fifth, and two 
fifths of a right angle ; i. e. 



, angles of 18°, etc., thus : 





Ex. 70. Divide a right angle into two parts, one of 40° the 
other of 50°. This can be most easily done by finding one-third 
of the right angle, and making the angle and arc of 40°, one-third 
greater than the one of 30°, thus : 




Ex. 71. Repeat the divisions of the right angle, given in the 



26 



FREE-HAND GEOMETRICAL DRAWING. 



preceding examples, upon right angles in various oblique posi 
tions as in Ex, 36. 

Practical Examples. 




Ex. 72. A four pointed star, requiring two lines at right 
angles to each other, and the equal bisecting lines of those 
angles. 




Ex. 73. A gate. JSTote that an angle of 24° is four-fif teentha 
of a right angle. 



£ 



r~^ 



Ex. 74. An arch, giving practice in parallels, equal distances 
(each side of the arch, and the heights at the ends) and arcs, of 
various sizes, and parts of a circle. 



CHAPTER VI. 

PLANE FIGURES BOUNDED BY STRAIGHT LINES. 

Principles. 

A. plane figure is a portion of a flat surface, bounded bylines 
When bounded by straight lines, it is called a polygon. 

Polygons are of various names, depending on their number 
of sides. 

A Triangle has the least possible number of sides, viz., three. 
It has also three angles, and when one of these is a right angle, 
the triangle is called right angled. 

A Quadrilateral, or quadrangle, has four sides, and angles. 
When both the angles and sides are equal, the figure is a square, 
and its angles are all equal. When the angles are right angles, 
but only the opposite sides are equal, the figure is called a rec- 
tangle. 

A Pe7itagon is a figure of five sides. In a regular pentagon 
the sides and angles are all equal. 

Likewise, a regular Hexagon has six equal sides and angles. 

The diagonal of a four-sided figure joins its opposite corners, 
thus : 




Figures of more than four sides, have more than one diagonal 
from any one corner. 

The student is now prepared to sketch such simple objects as 
depend only on certain proportions between their angles. 

According to the theory of beauty of angular proportions, 
briefly alluded to in Chapter II., those regular figures are most 
beautiful, in which the proportions of the angles can be ex- 
pressed by fractions whose terms are small numbers. 

A great many familiar objects have sides of an oblong, that 
is a rectangular form, and these sides are divided by their 
diagonals into two equal right angled triangles. A triangle is 
the simplest plane figure, and a right angled triangle is the 
simplest triangle, as a standard for the comparison of angular 



28 



FEEE-HAND GEOMETRICAL DRAWING. 



proportions, since it contains a right angle, which, is the natui al 
measure with which to compare its other angles. 

Rectangles, as floors, walls, doors, windows, the spaces be- 
tween them, etc., are therefore, most beautifully proportioned, 
when their diagonals divide their right angles into parts 
hearing a simple proportion to each other and to a right angle. 

Thus, if the diagonal of a rectangle divides one of its right 
angles into angles of 30° and 60°, the ratio of these is -J-, and 
their ratios to a right angle, are -J- and f . These all being sim- 
ple fractions, the rectangle will be found to have agreeable pro- 
portions. 

Elementary Exam/pies. 
The construction of regular figures, requires attention to the 
equality of some or all of the sides, as in Chapter III., as well 
as to their direction, and the proper size of their angles ; and 
thus requires the application of examples in all the preceding 
chapters. 





Ex. 75. A right 
angled triangle 
with equal acute 
each. 



angles of 45 c 



This triangle possesses the property of being divided by a per- 
pendicular from its right angle to its opposite side, into two 
triangles of the same shape as the original whole. This prop- 
erty makes its construction easy. Draw this triangle in various 
positions, and fill it with lines parallel to its longest side, as 
above. 




Ex. 76. A triangle each of whose halves is a right angled 



PLANE FIGURES BOUNDED BY STRAIGHT LINES. 



21) 



triangle with acute angles of 36° and 54°. Here ||o=| ; $ j:=| 
and $$o=f. Also in the whole triangle ■3£^,=$. These ratios 
being varied, while all of them are simple, the triangle is very 
pleasing and forms an agreeable end, or " pediment," to a roof, 
as seen in the figure. 

Ex. 77. An equal sided triangle. This also, has equal angles 
of 60° each, and its halves therefore have 
acute angles of 30° and 60°. Draw several 
such triangles, and fill each one of some of 
them with one or more sets of lines, parallel, 
or perpendicular, to some one of its sides. 




Ex. 78. Construct squares of various sizes and in various 
positions, first without their diagonals and then with them. 

Ex. 79. A figure of four equal sides, but whose opposite 
angles, only, are equal, is called a Rhombus, thus : 




This figure is most easily constructed by first drawing its 
diagonals so that each shall be at right angles to the other at its 
middle point, and by then joining their extremities. 

Let rhombuses of various proportions be drawn. 

A square may also be drawn by its diagonals in the same way. 

Ex. 80. After the practice thus far had, various designs in 
plane figures can be executed, such as the following. These 




■ i ; 


► O O — i 

k ik A. AK -A 



examples obviously require the divisions of lines into equal 



30 FREE-HAND GEOMETRICAL DRAWING. 

parts. Also, in the second figure, the marking of equal dis- 
tances, viz., the semi-diagonals of the little squares. 

Ex. 81. Embraces a regular pentagon and some applications 
of it. 





The external angles of a pentagon formed by producing or 
extending its sides, are each equal to 72°, or four-fifths of a right 
angle, and are constructed accordingly. The five pointed star 
is most agreeably proportioned, by joining the alternate points 
in order to obtain the direction of the sides of the star points. 
Also, the middle line of any point, when extended, becomes the 
dividing line between the two opposite points. 

Ex. 82. Hexagons. These polygons have angles of 120° at 
their corners. They can therefore be combined as in pavements, 
so as to completely fill a given space. It will assist in construct- 
ing this figure, to remember that each of its sides is equal to the 
distance from its corners to its centre. Observe, also, that the 



longer diagonal is divided into four equal parts by the shorter 
ones, perpendicular to it, and the centre. 

Ex. 83. Divide a circle into eight equal parts, by diameters 
at 45° with each other, and join the points of division by straight 
lines, which will give a regular octagon, or eight sided figure. 
This figure can also be drawn, by considering that its external 
angles are each equal to 45°, thus : 



PLANE FIGURES BOUNDED BY STRAIGHT LINES. 31 



Practical Examples. 
Ex. 84. Wholly made up of vertical and horizontal lines. 



£ 



a 



Ll 



d 



a. 



JJ 



mj 



3 



u 






W/////yy//////////////////////////////y//////y/^^ 

Ex. 85. Embraces oblique lines. 









32 FREE-HAND GEOMETRICAL DRAWING. 

Ex. 86. Embraces circular lines. 






The figures of interlacing lines are best made by sketching 
the whole in very faint, and lonbroken lines, at first ; after which 
those portions of each line, which are meant to be visible, can 
be retraced in firm and heavy strokes. 






IXk I 




■ ~.y, ; 

\ „ 

. , ; 






'Aa-y^A 




•! 




^*V 


i. 



5 






ft 



U : 



s 



-4. 




m 



PL. IX 



\ \ \, 


n 




\ \ \ 






\ \ \ 


a. 




\. \ c \ 






a ' a * ^ 


V 






\. 



IP^ — » 



§MM 




CHAPTER VII. 



RECTILINEAR AND CIRCULAR COMBINATIONS. 

Principles. 
We gain, from observation of ornamental figures, and notably 



from that 



entertaining 



instrument, the kaleidoscope, certain 
which we will call those of unity, symmetry, and variety, 
in connection with figures compounded of a greater or less 
number of simple elements. 

These ideas, all of which are pleasing, we will now proceed 
to explain, and illustrate. 

Unity is that property of a figure by which, although com- 
posed of parts, its parts are so linked together that it addresses 
the mind as .one figure, and not as a collection of separate 
things. Thus three detached lines are naturally regarded as 
three separate things ; but, as combined in a triangle, perfectly 
enclosing a space, they are naturally thought of as forming one 
thing, the triangle. So also, with the figures in the kaleido- 
scope, their regular arrangement around a single central point, 
gives them unity, so that we think of each as one figure. 

Symmetry is single, double, or multiple. 

A figure has single symmetry when it is divisible by only 
one line into two parts which will coincide 
when folded together about that line. Thus 
a butterfly has such symmetry, his wings co- 
inciding, when folded together about the 
centre line of his back. Also any triangle, 
two of whose sides are equal, has a single 
line of symmetry, as AC in the next figure. 

A figure has double symmetry, when it is 
divisible by two lines in the manner just de- 
scribed. Thus the rectangle ABCD, shown 
on next page, has two axes or lines of symmetry ah and cd. 

Figures having more than two such lines, or axes, of sym- 
metry, have multiple symmetry. Thus a square has four, two 




34 



FREE-HAND GEOMETRICAL DRAWING. 



of which are its diagonals, and a circle has an infinite number, 
all its diameters being its lines of symmetry. 



B 



By variety we here mean precisely what takes place in turn- 
ing the kaleidoscope, that is, the pleasing result of different 
combinations of the same given elements. 

The figures on plates III., IV., and V., will illustrate the ideas 
of unity, symmetry and variety, as here explained, aud will 
afford examples of combination, suggesting many others. 

Illustration. We have, first, eight different combinations of 
four equal right-angled triangles. 

The first, PL III. Fig. 1, has unity in the close union of the 
triangles, but lacks symmetry, and is thus less pleasing than 
Fig. 7 which possesses both, though having only single sym- 
metry. 

Fig. 3 is without symmetry and is weak in unity, its parts 
being only united at a point, and is of inferior beauty. 

Fig. 2 has double symmetry, but is weak in unity, the tri- 
angles being joined by their shorter sides. It is thus less pleas- 
ing than Fig. 4, where unity is mere strongly expressed, by the 
union of the longer sides. 

Figs. 5 and 6 both have double symmetry / Fig. 5 satisfies 
the idea of unity by means of its unbroken circumference, and 
Fig. t> does the same by its solid union of the triangles. Both 
are pleasing, as is more apparent in case of Fig. 6, by drawing 
it as in Fig. 8, where the double lines, marking intermediate 
bands between the triangles, add richness to the figure. 



Exercises. 
From the above full illustration, the learner can proceed to 



RECTILINEAR AND CIRCULAR COMBINATIONS. 35 

invent as many combinations as possible, of the following sets 
of figures. 

Ex. S7. Make various combinations of four equal acute 
isosceles triangles. See one in PL III., Fig. 9. 

Ex. 8S. Do the same with four equal obtuse isosceles tri- 
angles. See one in PL III., Fig. 10. 

Ex. 89. Do. with four scalene triangles. See one in PL 
III., Fig. 11. 

Ex. 90. Do. with four equal squares. See PL III., Fig. 
12, for one. 

Ex. 91. Do. with four equal rhombuses. See PL Y., 
Fig. 1. 

Ex. 92. Do. with four trapezoids ; a figure of single sym- 
metry with two parallel sides. See PL Y., Fig. 2. 

General Example. Vary the last six examples in one or 
more of the following ways : 

(1) By increasing the number of figures to be combined, still 
keeping them equal. 

(2) By making them unequal, but still similar, as in com- 
bining large and small rhombuses or trapezoids, as in PL Y., 
Fig. 3, of rhombuses, where the eight angles around the cen- 
tral point are equal. 

(3) By making the elementary figures unequal and dissimi- 
lar, or at least dissimilar, as in PL Y., Fig. 5, a pendant composed 
of right angled, acute angled, and obtuse angled triangles ; Fig. 
4 a decorative cross, and Fig. 6, of various rhombuses-like figures 
of single symmetry (or mono-symmetrical, they may be called). 

Fig. 5 may be supplemented by additions, like the one shown, 
at each of the three remaining corners of the square ; and may 
be varied by other arrangements of the pendant triangle, or by 
using other figures. 

Plate IY. shows other examples of symmetry and varied 
combination based upon a square foundation. 

Ex. 93. Fig. 1 shows a doubly symmetrical arrangement of 
four-sided figures of single symmetry. 

Ex. 94. Fig. 2 shows the variation of Fig. 1, by placing the 
right angles of its component figures at the centre. Both ar- 
rangements might alternate in the same figure ; which would 
then better be placed diagonally. 



36 FREE-HAND GEOMETRICAL DRAWING. 

Ex. 95. In Fig. 3, a combination of squares, the sides of 
each making angles of 45° with the next, the angles of each 
might have extended beyond the circumference of the next 
outer one. 

Ex. 96. Fig. 4 might be varied by many different ways of 
occupying the angles between the equal arms of the cross. 

Exs. 97—98. Figs 5, 6, each give but one-half of a figure of 
single symmetry, the whole figure to be drawn by the student. 

Exs. 99-101. Figs. 7, 8, 9, are mostly composed of circular 
elements, arranged upon a foundation circle, the first, a 
rosette of six points ; the second, having the centres of the 
looped arcs at the angles of the curved quadrangle ; the last, 
the centres of the outer arcs of any leaf, at the extremities of 
the diameter which passes through the next leaf. 

By means of the suggestions attached to these examples, the 
learner will readily find the path of discovery leading to new 
designs, as well as to other variations of those here given. 

The figures of PL IV., may well be drawn of such size that 
four, or even two of them will fill the plate. 

The use of the segments of centre lines, as in Figs. 2, 8, 9, 
serves to give vividness to the figures by emphasizing the quali- 
ties of unity and symmetry possessed by them. This will be- 
come more evident by omitting them, and then comparing 
the results, for the same figure. See also, Fig. 3, where the 
blank centre weakens the expression of unity. 

Ex. 102. Combine circles by interlacing, as in PL V., Fig. 
7. Also with their centres arranged on a circle. 



CHAPTER VIII. 
Curves and Curved Objects in General. 

We have thus far mostly considered circular curves. These, 
however, are only the simplest among an endless diversity of 
curves, many of which are of great beauty, as well as common 
usefulness. 

When any curve and straight line merely touch at one point, 
they are said to be tangent to each other, and just at the point 
of touch, or tangency, they lie in the same direction. Hence 
any curve can be much more easily sketched, if we know several 
tangents to it at different points. 

A circle can evidently be placed, or " inscribed" in a square, 
so as to be tangent to it at the middle point of each side. A 
curve similarly inscribed in a rectangle is called an ellipse. 
Kow observe that as all squares are of the same shape, though 
of different sizes, so all circles must be of the same shape, also. 
But there is an endless variety in the proportions of different 
rectangles, and hence there may be an equal variety of ellipses. 

A right angle being more easily estimated than other angles. 
it is also a special help, in sketching a curve, to have one or more 
lines which the curve must cross at right angles. Hence it will 
be easier to sketch an ellipse in a rhombus than in a rectangle ; 
for in the former, the ellipse will be tangent to the four sides, 
and will cross each diagonal, at right angles with it, and at 
equal distances from its extremities. 

Ex. 103. Sketch ellipses of various proportions by the rhom- 
boiclal method, thus : Mark the middle point of each side, as 





points of tangency of the ellipse; then, make each diagonal of 
the rhombus, the diagonal of a square containing an inscribed 



38 



FREE-HAND GEOMETRICAL DRAWING. 



circle, which will cross those diagonals at other points of the 
ellipse. 

Let this exercise be continued, in the sketching of ellipses 
in rhombuses placed in various oblique positions, and, also, with 
their longer diagonals placed vertically. 

"When an ellipse is inscribed in a rectangle, it crosses the cen- 
tre lines of the rectangle at right angles, at the points of tan- 
gency with the sides of the rectangle. Thus the eight guiding 
positions afforded by the rhombus, are reduced by union to four, 
in the rectangle. The ellipse will, however, cross the diagonals 
of a rectangle at equal distances from its corners, but not in a 
perpendicular direction. 

Ex. 104. Sketch ellipses in rectangles and other figures, of 
various proportions and positions, thus : 




An ellipse is a curve of most delicate grace, and should there- 
fore be most faithfully studied and carefully drawn. The most 
offensive error in shaping it, is, to represent it as pointed at the 
narrow end, which it is not, in the least. 

By combining elliptical arcs of various proportions, tangent to 
each other, various graceful forms adapted to ornaments, such 
as vases, may be composed. In doing this the relative propor- 
tions of the ellipses should not be chosen at random, but so 
that the angles of their enclosing rhombuses should form simple 
ratios. Moreover, these rhombuses should be in simple relative 
positions, and the corresponding angles in the different ones 
should form simple ratios. 

Ex. 105. In this design for a vase, all the angles, some of whose 
degrees are given in the enclosed numbers, are 9°, the square 
of 9°, or even multiples of 9°. Also at the base, two rhom- 
buses have a common vertex ; and at top, two have a side and t\. o 



CURVES AND CURVED OBJECTS IN GENERAL 



3i> 




vertices in common. The 

acute rim-rhombus has its 

sides perpendicular to a 54 

and b 72, its right side 

passes through the corner 72, 

and its diagonal passes 

through c, the junction of 

two arcs, and centre of a 

72. Moreover, the diagonal 

72-36 coincides with 18-72 

produced, and the side 72- 

54 is parallel to the diagon- 
al 18-36. 

These mostly very simple 

relations of the rhombuses, 

and their angles, yield a very 

pleasing form, each side of 

which embraces four differ- 
ent elliptical arcs, of which 

the one running upward from c terminates on a 54. 

Ex. 106. In this design, the relations 
are in part, more, and in part less simple 
than in the preceding, and the result will 
hardly be thought more agreeable than 
before. 

The principal, and the base rhombuses 
are of the same proportions, as seen by 
their angles, and therefore enclose simi- 
lar ellipses, which gives less decided vari- 
ety in the outline at the base. The up- 
per side-rhombus, with its angle of 18°, 
side of one in the central rhombus of 
60 D , gives the comparatively complex 
and unfamiliar ratio ■£$. Also its right 
hand corner is arbitrarily located on a 

horizontal line through the upper vertex of the central rhombus. 
In both of these designs the rolling rim might be omitted by 

terminating the sides of the vases on the longer diagonals of the 

narrow upper side rectangles. 

Ex. 107. By substituting for a rhombus, two dissimilar half 




40 



FREE-HAND GEOMETRICAL DRAWING. 



rhombuses, having a diagonal in common, the beautiful egg 
shaped curve will be formed, thus: 





In the first of these figures, the acute half angles are 20° and 
30°, whose ratio is therefore f. In the second figure the cor- 
responding angles are 18° and 36°, having therefore a ratio of 
J, and affording a more decidedly egg-shaped curve. 

Ex. 108. An egg-shaped oval may also be inscribed in a 
regular trapezoid, that is a figure having two unequal but paral- 
lel sides, both of which are bisected by the same line, perpen- 
dicular to both, thus : 





Let these ovals be drawn in a great variety of proportions and 
positions, both in rhombus-like figures and trapezoids, and with 
as frequent reference as possible to leaves, which exhibit a great 
variety of graceful ovals. 



CURVES AND CURVED OBJECTS IN GENERAL. 



41 



Ex. 109. The material of vases, etc., being ^ 
originally plastic, it may be supposed to settle 
by its own weight into oval forms before har- 
dening. For this reason, as well as from the 
greater stability associated with breadth at 
base, egg forms are more admired in pottery 
articles than true ellipses. The annexed de- 
sign illustrates these remarks. Its angles of 
36° and 54° ; 54° and 10°-48' (ten degrees 
and forty-eight minutes) 75°-36' and 18°-54', 
give the simple ratios -f, -J-, i, J. The student 
should make a variety of similar designs. 

Examples 105, 106, 109, are here incidental 
and preliminary, illustrating a manner of 
using ellipses and ovals by means of circum- 
scribed rhombuses, etc. The use of ovals in 
designs will be more fully and systematically 
explained in Part III. 

Ex. 110. On account of the pleasing associations of stability 
and decision with horizontal and vertical lines, as indicated in 
Chap. L, a curve which enters into the composition of any solid 

and fixed object is most pleasing 
when it has one or more horizon- 
tal or vertical tangents. 

Thus, there is more vigor, as 
well as variety, in the curve in 
the second of these figures, than 
in the first. 

Ex. 111. As we here propose only such exercises as are more 
closely associated with geometrical drawing, we only allude to 







the careful drawing of German text and common writing (script) 



42 FREE-HAND GEOMETRICAL DRAWING. 

letters on a large scale, as an excellent exercise in the close 
study and varied practice of drawing curves. 





The German text, and all upright letters should be evenly 
balanced on each side of an imaginary vertical centre line, in 
order to give them the most satisfactory appearance. 

Ex. 112. The varieties of curves being innumerable, a few are 
here annexed by way of suggestion. The student can devise 
many others. 




The group of four parallel curves affords an excellent ex- 
ample for practice, each curve being nearly straight in the mid- 
dle, and sharply curved at the ends, while its left-hand half 
is convex upwards, and its right-hand half equally so down- 
wards ; and each with a vertical tangent at its extremities. 

Of the two spirals, it will be seen that one increases its radius 
uniformly, giving equal radial distances between its successive 
turns, while the other expands at an increasing rate. 



CURVES AND CURVED OBJECTS IN GENERAL. 



43 



The vise of tangents in sketching curves is also illustrated in 
these examples. 




Ex. 113. An exercise of peculiar utility, is found in sketching 
easy curves through several given scattered points. This opera- 
tion frequently occurs in geometrical drawing, when other than 






circular curves are to be described. The essential things to be 
observed in these cases, are, first, to avoid all sudden, irregular, 
and unnecessary variations in the rate or degree of curvature, 



44 



FREE- HAND GEOMETRICAL DRAWING. 



and, second, especially to avoid making an angle at any point 
in the intended curve. These important requirements can be 
met by keeping at least three successive points in view at once. 
Thus, while joining A and B in the figure, keep C in view, and 
operate likewise in making all similar figures. 

The student should practice extensively on this example, 
first talcing the points, in many different relative positions, and 
then running easily flowing curves through them. 

Ex. 114. In several of the preceding examples, curves have 
been drawn tangent to straight lines previously drawn. We 
here add an example of drawing tangents to curves already 
drawn. 





The tangent may be drawn through a given point out of the 
curve, as in the first figure, or through a given point on the 
curve as in the second figure. 

Ex. 115. Finally, the examples of this chapter close with 
practice in the very nice operation of drawing symmetrical 
figures with variously curved outlines. Symmetrical figures 
(p. 33), are those which are divided in one or more ways, by a 
centre line, into similar halves, as in this figure. The difficulty 
in such figures, after forming one side in a pleasing curve, is, to 
make the other side of exactly the same form, but in a reversed 
position. This can be done, as in the 
figure, by drawing lines perpendicular 
to the centre line, and by marking on 
them equal distances on each side of 
the centre line. The following are 
other examples of symmetrical figures, 
some of which have two centre lines. 
The learner can devise many other 
figures of similar character. 

Ex. 116. Let PL IV., Figs. 10, 11, 
each be taken as one half of a sym- 
metrical figure, and draw the other half. 




CURVES AND CURVED OBJECTS IN GENERAL. 45 





46 FREE-HAND GEOMETRICAL DRAWING. 

Ex. 117. Representing a few elementary corner pieces, illus- 
trates some of the foregoing principles. 10 is inferior to 9 be- 
cause its main spur seems weakly placed, or driven in, while the 
spurred corner, 9, is firmly planted. 7 is 
better than 6, because it cuts out less of 
the interior, and because the grace of 
the curve is protected by the strength 
of the square corners at each side of it. 
Thus the skeleton of every corner 
should embody a good idea, for no rich- 
ness of detail in ornament can redeem 
bad governing outlines. 
Attention to such simple principles as these will guide in the 
design or selection of borders, and prevent the necessity of pre- 
senting an elaborate collection of them here, when they can be 
seen in such abundance in type-founders' collections, and in 
engravers' and printers' works, together with various orna- 
mental devices. 

Another principle, disregard of which through dispropor- 
tionate interest in some trivial thing, may spoil a good drawing, 
is this. Ornamental devices on drawings of solid worth, should 
never represent anything essentially mean, or rudely comic, or 
even anything of merely transient interest. Neither should they 
be attempted uuless they are sure to be well executed. 

Thus, a vignette on a map of a survey may contain a sketch 
of some pretty view seen from some point. A drawing of an 
engineering structure or a machine, may contain a pictorial 
view of the same object; or of the establishment where it was 
made, or of the room or building in which the drawing was 
made ; anything in short, which is agreeable in itself, and not 
foreign to the subject. 

Ex. 118. As a concluding exercise upon useful symmetrical 
figures, various ornamental arrow, or spear heads may be drawn. 
See PI. Y., Fig. 8. These are useful in iron-work, and as devices 
for vanes, and as indicating the meridian in maps of surveys. 






PL. 'Ill, 








m. 



P.L. IV. 




PL.V. 




CHAPTER IX. 

LETTERING. 

General Principles. 

Lettering, though not strictly a part of a drawing, is a neces- 
sary appendage to it, it being generally indispensable to the full 
understanding and intended use of the drawing. And as, also, 
there should be uniformity of accuracy and elegance in all parts 
of the draftsman's work, lettering is properly included among the 
fundamental operations, which he should be familiar w T ith 
before applying his art in practical cases. 

Besides, although geometrical drawings should be principally 
titled with geometrical letters, yet these letters are, on account 
of their usually moderate size, as well as variety and curvature 
of outline, most conveniently made by the free hand. Hence 
the draftsman's training in lettering appropriately falls among 
the subjects of free geometrical drawing. 

Two points should be constantly remembered during the 
practice of lettering : first, uniformity of size and proportions, 
and, second, beauty and regularity of form in each letter. Ill- 
shaped letters, if of uniform size, proportions, and distance apart, 
and truly ranged in a straight line or regular curve, w r ill look 
tolerably neat. Elegant letters will, on the other hand, appeal 
badly, if irregularly sized and located. Both uniformity, and 
elegance are, therefore, indispensable to perfect lettering. 

The learner's previous practice, in marking equal and propor- 
tional distances and angles, should enable him to secure uni- 
formity in his letters ; and his practice on curved and other 
irregular lines and figures, should enable him to give them 
elegance of form. 

All the letters described in this chapter should "first be drawn 
on plates of smooth heavy brown paper, about 11 by 14 inches 
in size, and with a crayon or soft pencil. They should be made 
three or four inches high, so as to afford exercise in free and 
broad movements of the hand, and may afterwards be made of 
ordinary sizes, on smaller plates, and in title pages. 



48 FREE-HAND GEOMETRICAL DRAWING. 

Roman Capitals. 

Before entering upon a general discussion of all the varieties 
of letters, we will make a special study of the common Roman 
capital letter, which is a sort of standard which all other letters 
are made to resemble, more or less closely, in certain particulars ; 
and from which, as a starting point, variations are made in de- 
signing fanciful letters. 

Plate VI. The Alphabet in Large Roman Capitals. — This 
alphabet is arranged in three groups, so as to form progressive 
exercises in the drawing of the letters. The first group em- 
braces those letters such as I and IT, etc., which are composed, 
wholly or mostly, of horizontal and vertical straight lines. The 
second group contains all those letters in which oblique straight 
lines are prominent ; while the third group embraces those let- 
ters which are largely made up of curved lines. 

Letters, as large as those of this plate, may be made by in- 
struments, by observing certain proportions in their form ; but, 
inasmuch as, in common practice, letters are of such size that 
they are more conveniently made by hand, it will be far better 
for the student to make the large letters* of PL VI. by hand, 
at least so far as to sketch their curved lines, and the points 
through which their straight lines pass ; after which, the lines, 
if inked, may be ruled. A running commentary on the different 
letters of PL VI. will now be sufficient. I, the simplest of all 
the letters, consists of a vertical column, whose width may pro- 
perly be made equal to a quarter of its entire height. The caps 
at the top and bottom project beyond the column a distance on 
each side, equal to half the width of the column. These pro- 
portions may be observed in the wide parts and caps of all the 
letters. 

"We thus have for an I the following complete proportions : 
Divide its height into sixteen equal parts. Then its height = 
f|, its total width T 8 g, width of column ^, projection of cap -&, 
and thickness of cap -^. These dimensions are to be preserved 
in the vertical columns of all the letters. Also all wide columns 
are to be of -^perpendicular width, and all the caps are to be 
^ig- thick. 

Having thus fixed upon a proper thickness for the caps, let 
lines be ruled parallel to the extreme top and bottom lines, to 
aid in making these caps of uniform thickness on all the letters 



LETTERING. 49 

Each column of the II is like an I. The extreme width of 
this letter allowing T \ between the caps is equal to {% of its total 
height. 

The height of the arm of the L is -^ of the total height of the 
letter. The extreme width of this letter, and of F, making the 
arms -^ longer than they are high, is {£ of the height. The 
ends of the arms must be ^ thick. F is like an L turned up- 
side down, with the addition of the middle arm, whose height 
is half the height of the letter, and whose right-hand line is 
midway between the right-hand line of the column and the ex- 
treme right-hand line of the letter. E differs from F only in 
having another arm. Some designers make this letter a little 
wider (|f) at bottom than at the top, and also make the height 
of the top arm a little less than that of the lower one. This 
method gives variety and an appearance of stability. 

T, having an arm on each side of a central column, has its 
extreme width equal -}-§- of its total height. Notice, on all these 
arms, that their curved sides are nearly quarter circles, giving 
solidity of appearance to the arms. None of these arms should 
be short, thin, or pointed. 

Passing the hyphen we come to letters having oblique ele- 
ments. V having its average width only equal to half its ex- 
treme width, since it comes nearly to a point at one extremity, 
may be made of extra width at the top ; thus, let the total width 
be such as would be given by two wide columns with -^ between 
their caps. This width will then be \$* of the whole height 
Let the perpendicular width of all narrow columns be ^ and 
the horizontal width of V and A at their points -^. 

Observe, that the left hand column is the wide one, and 
that in all letters having slanting columns, except Z, the heavy 
column slants downward towards the right: Similar general 
directions to the preceding, apply to A. The cross bar of this 
letter may be half way from the bottom line to the inner angle. 
In K the under side of the narrow arm may intersect the ver- 
tical column, a little below the middle, as at two-fifths of its 
height, so that the wide oblique column may not intersect the 
vertical column. The extreme width at the top equals the total 
height, and at the bottom equals Sffi of the whole height. 

N", having an oblique wide column, but being a square letter, 
having two vertical columns, does not need the extra width given 



50 FREE-HAND GEOMETRICAL DRAWING. 

to Y and A. The length of full caps to oblique wide columns 
being f-f , and to vertical narrow ones -j^-, the total width at top. 
allowing y^- between caps, if there were a full cap at the left 
upper corner, will be *£$. There is no cap at the lower right- 
hand corner. 

The under edge of its wide column is drawn from the left 
side of the foot of the right-hand narrow column, tangent to 
the slight curve which connects the upper left-hand cap with 
the left-hand narrow column. M has its total width equal to 
W of its total height. The point of the Y-shaped part is on the 
bottom line, and midway between the inner lines of the adja- 
cent vertical columns. W, the widest letter of the alphabet, 
is of an extreme width equal to 3 T 7 ^ of its extreme height. Its 
oblique lines are parallel to the corresponding lines of Y. The 
extreme width of Z is equal to ^^ of the height. Its arms are 
lengthened, as there are no caps opposite to them. The lower 
one is |-§- long and T 8 g- high, the upper T ^- long and -^ high. 

The left-hand vertical lines of the left-hand caps of X are 
in a vertical line. Reckoning from these lines, the extreme 
width at bottom is equal to l -ff of the total height, and at 
top it is equal to -^f- of the height. In Y the outer oblique 
lines intersect the vertical column a little below the middle, as 
at a distance equal to the thickness of the caps. The whole 
width at the top equals -^J of the whole height. 

Passing the second hyphen, we come now to letters in which 
curves form a prominent part. The total width of J is -^J- of its 
height. Its larger curve, convex downward, has for a chord a hori- 
zontal line, at a height above the bottom equal to -f-^ of the height. 
The extreme width of U is jf , of D f£, of P -if, and B ||, of the 
height. The bow of the P should intersect the column a little 
below the middle, while the upper bow of the B may properly 
intersect the column a little above the middle, making the lower 
bow project y 1 ^- beyond the upper one. P is if wide at bottom. 
It differs from B so little, as not to need further description. By 
omitting the tail of the Q it becomes an O. The greatest width 
of the tail equals that of a wide column, and it extends three- 
fourths of the same width below the body of the letter. In 
either case the extreme width equals T f of the height. The 
extreme width of C equals -J-J-. The highest and lowest points of 
its outer curve are in the middle of the extreme width ; and the 



LETTERING. 51 

corresponding points of the inner curve are half way between 
the inner point of the lower curved arm and the vertical tangent 
to the inner curve. In a letter as large as this, it is w r ell to let 
the upper arm set back, a distance equal to the thicknes of a 
cap, so as to prevent the overhanging look that it otherwise 
would have. The extreme width of Gr equals its total height. 
Its construction is evident from the figure, after the description 
of 0, that has been given. The whole width of S equals that of 
Q, and its arms are nearly like those of C. Some designers 
make the lower half higher and wider than the upper half, but 
as S is, to a beginner, the most troublesome letter, it is here 
given in its simplest form. To sketch it readily, it is only neces- 
sary to keep in mind that the outer curve at the top becomes 
the inner one in the lower half, and so must be carried below 
the middle of the letter and curved sharply to form the inner 
line of the lower half. & is less subject to rule than the proper 
letters of the alphabet. The design on PL I. is offered as being 
more pleasing than that in which the wing over the period 
ends in a rectangular cap. The dotted lines show a modifica- 
tion of the design, ending in a large circle. 

The proportions here given are not absolute, but only relative. 
Thus an ordinary letter, as an IT, or an E, may be made twice as 
wide as it is high, or half as wide as it is high, but in that case 
all the other letters would have similar modifications of their 
present proportions. Such letters are called, respectively, ex- 
panded and condensed letters. 

Directions, much more minute than the preceding, are some- 
times given for lettering, but, after affording a few essential hints 
concerning the general proportions of letters, it is here preferred 
to leave the details of their design to the taste and judgment of 
the designer. 

Letters in General. 

In examining a type-founder's specimen book, one ma} r 
imagine, from the exceeding variety of letters therein exhibited, 
that it must be impossible to reduce them to any system. But 
a closer examination will reveal a few comprehensive features, 
according to which all letters may be readily classified in 
groups. 



52 FREE-HAND GEOMETRICAL DRAWING. 

By acquaintance with the distinguishing characters of these 
groups, and their modes of variation from one another, it will 
be easy to design uniform letters in any proposed form or style, 
which is much better than a mere copying of them, without 
ability to proceed independently of a copy. 

All letters may be included in two grand divisions. 

I. — Geometrical letters are all those which have a definite 
geometrical outline, which, when sufficiently large, could be 
made with drawing instruments ; and — 

II. — Free-hand letters, or those of so irregularly varied out- 
line that they must be made by hand only, guided mainly by 
the fancy of the designer. 

Since the letters called geometrical are the ones mainly used 
in geometrical drawing, they will chiefly be noticed in this 
section. The student, by collecting a number of hand-bills, 
programmes, business cards, sheet-music covers, etc., will have 
materials for a valuable scrap-book of letters, which will be 
useful for reference, and will contain numerous practical illus- 
trations of the explanations which follow. 

By examining such a collection, it will be seen that in all or- 
dinary letters three things may be distinguished — 

(a) the essential elements. 

(b) the complementary additions. 

(c) the decorations. 

The essential elements of letters, are those which are neces- 
sary, and sufficient, to enable one to recognize the letters. The 
first half of the first and third lines, and the second and fourth 
examples on the second line of PL TIL, are letters formed of 
essential elements only. 

The complementary additions are the caps, and the hanging 
parts of the arms, etc. The letters of the first five, and the 
seventh, lines of PL VII. are, with the exception of those just 
mentioned, letters having these additions. 

By the decorations are meant the ornamental shading and 
filling up of the letters. Thus letters may be represented as if 
made of wood, stone, or iron ; and of pieces having square or 
polygonal sections. They may appear as if seen obliquely, or 
as draped, vine-clad, or casting shadows. 

In spacing letters, it is a good rule to allow equal areas of 
blank paper between them. 



LETTERING. 



53 



Summing up; the preceding, and other particulars concern- 
ing letters, are systematically presented in the following table : — 



o r— :a O 



.a . 

Is 
a ?r 

If 



a « 



M !> GQ 



fl 



H 




<T3 © 

£ p< 

5 .9 




a 



I 

1 



.2 



m 



< 

g 
| 

§ a 



£ 5 



S.S 

rt3 



g 






a a 



H 



■i a 



.a 



! 

Pi 

a 



g 1-a 



i i 02 «-< 

Jh 52 fn s 

<u 3 <x> .t: 

p=h a -43^ 



' o 

2~8 



aav saaxxai 



54 FREE-HAND GEOMETRICAL DRAWING. 

It follows from this, that there cannot be very many radi- 
cally different forms of letters ; therefore, instead of making 
a further subdivision of geometrical letters, some of the ways 
may be mentioned in which varieties of letters are produced 
by modifications of the elements just given. 

1°. By altering the proportions of height and w T idth, forming 
expanded or condensed letters. 

2°. By retaining or omitting the complementary additions. 

3°. By making the wide columns of the letter massive or 
slender. 

4°. By making the letters as if they were flat plates, or as if 
they w T ere solid, or " block " letters. 

5°. By representing the latter as seen directly, or obliquely, 
so as to show both face and thickness. 

6°. By minor modifications in the outlines, as by rounding 
the caps into the columns. 

7°. By making the usually curved letters polygonal. 

8°. Varieties, without limit, may be made, by changes in the 
quantity and character of the decorations. 

Practical Remarks. 

{a) The thickness of the caps is the same as that of the nar- 
row essential elements. 

(b.) In pencilling letters, never pencil the ornaments, unless 
the letters are of extraordinary size, but pencil the outlines 
only, in very fine lines. 

(c.) It is better to do all the pencilling by hand, since instru- 
ments would perpetually be hiding portions of the letters, and 
so preventing the eye from judging readily of their proper pro- 
portions. 

(d.) Very small capitals and small letters are better put in 
off hand, in ink, between parallel pencil lines, to keep them of 
a uniform height. 

(e.) The sixth row of PL IV. shows a simple free-hand or 
" rustic" letter, in two sizes and styles. These are bark letters. 
Log letters are often seen in handbills, etc. 

(f.) The sixth row embraces " skeleton" and " full faced " 
" small " Roman letters and italics. A common error consists 
in making the stems of the b'syjp's, etc., too long. The total 



LETTERING. 55 

height of such letters need not be more than one and a half 
times the height of their bodies. 

(</.) To avoid making letters slightly leaning, stand directly 
in front of the work, and with the eyes far enough from the 
paper to be able to see the position of the border of the plate, 
a> a guide. Or, rule vertical parallels at short intervals. 

(/i.) Curves can be more neatly sketched in by a dotting, or 
very light motion of the pencil, than by a continuous motion 
with firm pressure. 

(i.) The ends of the arms of letters like G, C, S, etc., should 
not be far apart, vertically, but should come nearly together, 
and should be tangent to vertical lines, in order to give them a 
plump, finished, square, and stable look. 

(j.) Even in the most fanciful letters, there is a certain ap- 
preciable consistency and orderly form. This results from 
their having an imaginary central skeleton of regular single 
lines, about which the outlines of their parts are equally 
balanced. 

{7c.) PI. VII. illustrates most of the distinctions of form men- 
tioned in the preceding table, except the inelegant and unused 
Italian type. This plate, or one of similar nature, should be 
constructed by the student. 

(I.) Polygonal letters may be substituted for curved ones by 
any who are particularly deficient in free-hand sketching. They 
may thus be able to secure a desirable uniformity of excellence 
in their work ; though it is probable that the pains necessary 
to form an elegant polygonal letter would secure an equally 
elegant curved one. 

(m.) In line 2d, example 1st is elegant in being slightly ex- 
panded, and not heavy. Ex. 2d is neat and easy to make. In 
titles, the letters should be farther apart. Ex. 3d, of this line 
and of line 7th, are partly Italian in character, the essential parts 
being lightest. Ex. 4:th, is like Ex. 1st, line 1st, but heavier. 

In line 7th, Ex. 1st, is condensed ; Ex. 2d, shaded, and Ex. 
3d, spurred. 

(n.) After the systematic explanation of letter drawing, with 
varied illustrations, now given, the student should make an en- 
tire alphabet of each of the kinds of letters shown on PI. YII. 

(o.) In combining words to form the titles to maps and draw- 
ings, the most essential principles are the following. 



56 FREE-HAND GEOMETRICAL DRAWING. 

1°. To vary the letters in size, or heaviness, or both, accord 
ing to the relative importance of the different words of the 
title ; and the heaviness, or blackness, also, according to the 
general depth of color of the drawing. 

2°. To make decided contrasts between the lengths of the 
different lines of lettering contained in the title, and so that the 
circumscribing figure, formed by joining the ends of the suc- 
cessive lines, shall have a pleasing outline. 

When the spelling of the words makes this difficult, the use of 
" condensed," or " expanded " letters, or the prolongation of 
some of the lines of the title by long dashes, will afford con- 
siderable aid. 

In connection with these principles, study the artistic features 
of titles, and title pages, critically, with a view to good typo- 
graphical design. 

(jp.) In the old-fashioned styles of type recently revived, one 
of the most obvious marks of distinction is, that the arms of 
the E's, T's, etc., instead of being vertical, are divergent, as in 
the following letters. 

E F H L T Z 

(q.) Finally, remark (i) may be modified by adding that the 
arms should have vertical tangents, unless plainly meant not to 
have them, as if C, for example, in the first half of line 3rd, of 
PI. VII., were to consist of three-fourths of a full circle. 



PART II. 

SOLID DRA-WHSTG-, 



CHAPTER I. 



OBJECT, OR MODEL DRAWING. 

§ 1. — Rectilinear Models. 

Definitions and Principles. 

The angle made by two intersecting straight lines is called a 
plane angle. 

The angular space enclosed by three or more plane surfaces 
which meet at a point, is called a solid angle. Thus the angle 
at one of the upper corners, B, of a room, see the figure, 
where two walls, ABD and CBD, and the ceiling ABC meet, 





is a solid angle. In a square-cornered room, such an angle 
is a solid right angle. This is the simplest of solid angles, 
and is bounded by three plane right angles, one, ABC, in the 
ceiling formed by the meeting of two of its edges AB and 
3* 



58 FREE-HAND GEOMETRICAL DRAWING. 

CB, and one in each of the connected walls, as ABD and 
CBD. 

But there are many other solid angles, bounded by three or 
more plane angles, some or all of which may not be right 
angles. Thus the angle at the summit of any pyramid is a 
solid angle and is bounded by as many plane angles as the 
pyramid has sides. 

These simple principles being apprehended, no large mis- 
cellaneous collection of models is necessary in order to obtain 
skill in making free-hand sketches of geometrical objects from 
the solids themselves. A few variously proportioned prisms 
and pyramids each placed in various positions, and singly, or 
combined, will afford an almost unlimited variety of practice 
in combinations of length and direction of straight lines. 
These objects can be made by any wood- worker, or by pupils 
for themselves, and may usefully be of wood or pasteboard ; 
also skeleton forms may be made of wire or light wooden 
rods.* 

A set of simple plane or flat-sided drawing models, of con- 
venient size to be seen across a room, being provided, exercises 
upon them may be made suitably progressive according to the 
three following principles. 

I. Selecting for reference any one of the various solid angles 
of the body, let the position of the body be taken, first, so as to 
show only one of its bounding plane angles ; then two, and so 
on till the body shall be so placed as to show all the bounding 
plane angles of the given solid angle. 

II. Attending to the surfaces of the body,\et it first be placed 
so that but one such surface shall be visible, then two, and so 
on till the greatest possible number shall be visible. 

III. For each position of the solid or closed model, place 
by the side of it, and in the same position, a skeleton m,odel of 
the same body, which will further vary the exercise by showing 
lines that are hidden on the opaque model. 



* When it is desired to purchase manufactured models, Harding's English 
models, architectural in character, and affording many combinations, may be 
found useful. So, also, will the excellent elementary sets manufactured at 
the Worcester (Mass. ) Institute of Industrial Science, and which more closely 
agree with the principles here stated in the text. Other sets may perhaps bo 
found on inquiry at Art, or School Supply stores. 



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59 



Exercises. 

Ex. 119, PL VIII., Fig. 1, represents a cube, selected for 
illustration, and placed so that but one face is seen. 

Ex. 120, Fig. 2, represents the skeleton cube in the same 
position. All the edges will be visible, but in various ways, 
according as the eve of the observer is above or below, to the 
right or left of the position indicated by the figure, which is 
directly in front of E, the centre of the front of the cube. 

Ex. 121,. In Fig. 3, the eye is supposed to be above the 
level of the top of the cube, which is represented as turned so 
that three faces, c, m, and ??, of the same solid angle are 
seen. 

Ex. 122. Fig. 4 represents a similar position of the skele- 
ton cube. 

Ex. 123. Fig. 5 represents the cube as so related to the 
eye, that only e and n of the three angles shown in Fig. 3 are 
visible. The eye is here between the levels of the upper and 
lower bases of the cube. 

Ex. 124. Fig. 6 represents the skeleton cube situated as in 
Fig. 5. 

The pupil can easily satisfy himself by experiment that the 
nearer he approaches the cube, the more rapidly will the reced- 
ing lines as c d and ef, Fig. 3, appear to converge, and that the 
contrary result will take place, the further he removes from the 
object. 

Exercises like the foregoing may be arranged for each of the 
elementary solids. Thus : 

Ex. 125. Construct a series of six figures, of a solid, and of 
a skeleton triangular pyramid. 

Ex. 126. Do. of a square pyramid. 

Ex. 127. Do. of a hexagonal pyramid. 

Ex. 128. Do. of an octagonal pyramid. 

In each of these four exercises, it may be a solid angle at 
some corner of the base, or the one at the vertex which is 
chosen, for the purpose of guiding the changes of position of 
the pyramid. Thus, in the two following figures, Fig. 1 rep- 
resents a square pyramid, in which three sides of the solid 
angle at a are visible. The pyramid is thus either supposed to 



60 FREE-HAND GEOMETRICAL DRAWING. 

be tipped backward so as to bring its base in sight ; or else, if 
it stands on a level, to be above the level of the eye which may 
then be directly in front of some point as E. Fig. 2 represents 
a triangular pyramid, so placed that all of the plane angles at 
its vertex are visible. 





Proceeding with these exercises : 

Ex. 129. Draw six figures, three of a solid, and three of a 
skeleton triangular prism. 

In this, and in each of the following exercises, the prism 
may be successively placed so as to show one, two, or all of the 
plane angles which bound some one of the solid angles at the 
lower base ; or, to show the same for some one of the solid 
angles at the upper base. 

Ex. 130. Draw, as above, a solid and a skeleton square 
prism, each in three or more successive positions of increasing 
complexity. 

Ex. 131. Do the same with a hexagonal prism. 

Ex. 132. Do the same with an octagonal prism. 



§ 2. — Curvilinear Models. 

The three elementary round bodies are the cylinder, the 
cone, and the sphere. 

The sphere can never appear otherwise than circular, henc( 
not much practice upon it is required, with respect to its form ) 
though, in studying effects of light and shade, a series of variec 
exercises may be founded upon each body, by placing it in tht 
same position in various lights, as well as in different post 
tions in the same light. 



61 

The cylinder and cone can be placed in positions similar to 
those already described for the prism and pyramid. That is, 
with the upper base (or the vertex) inclined directly towards, 
or from the eye, inclined to the right or left or inclined diagon- 
ally forward or backward ; that is to the north-east, sonth-east, 
north-west, or south-west, the observer facing the north. 

Not more than half of the curved surface of a cylinder can 
be seen ; but the entire convex surface of a cone can be made 
visible by inclining its vertex sufficiently towards the eye. 

As the exercises proposed in this Second Part are intended 
to be drawn directly from objects conveniently placed before 
the eye, without the intervention of drawn or printed copies,* 
no engraved copies are here given ; but the pupil should draw 
next, each of the three round bodies in various positions. 

After sufficient practice in drawing single bodies, a final 
series of exercises should consist in drawing various combina- 
tions of them, the combinations consisting of different bodies 
of the same kind, as prisms ; and then of bodies of different 
kinds, as, for example, a group consisting of a prism, a pyra- 
mid, a cone, and a sphere, resting upon each other. 



* If, however, an intermediate stage of work be, in some cases, preferred, 
large line, or shaded copies of drawings of solids can doubtless be readily 
provided. 



CHAPTEK II. 

PERSPECTIVE AND PROJECTION FREE-HAND DRAWING. 

Definitions. 

J. A perspective drawing is one which represents the ob- 
ject as it would appear when seen at ordinary distances. The 
most obvious and familiar characteristic of such drawings is, 
that lines in them, whose originals on the object are parallel, 
generally converge to a common point. 

Thus, in looking down a long stretch of straight railroad 
track, or through a very long building, as a large freight depot, 
the rails in the one case, and the side walls in the other, appear 
to approach each other as they recede from the eye. 

For tliis reason, a perspective drawing is in one sense a dis- 
torted representation of the object drawn. That is, it does not 
represent it as the object actually is, but only as it appears, 
when viewed from some given point. For instance, see the 
six figures of the cube on PL VIII., the necessary convergence 
of the receding lines distorts the angles, making those which in 
reality are all right angles, appear either as acute or obtuse 
angles, according to their position relative to the eye of the ob- 
server. 

If, however, these oblique angles of the drawing impress the 
mind as being true representations of actual right angles on the 
object, this result is owing to the modifying effect of our in- 
timate knowledge of the actual shape of the familiar object. 
But note, that this effect will not be produced, unless the picture 
and the object are both viewed similarly, in respect to distance 
and direction. 

II. Projection drawings. When, however, the beholder is 
at a very great distance from the object, as compared with the 
dimensions of the latter, the lines which are parallel on the 
object, will be so on the drawing also. Drawings made under 
this supposition, viz., that the object is seen from an inde- 



PERSPECTIVE AND PROJECTION FREE-HAND DRAWING. 63 

finitely great distance, are called projection drawings, or 
simply projections. 

Both of the kinds of drawing just described, and whether 
made by the free hand or instruinentally, are of use to every 
one who has occasion to draw. Yet as perspective drawings 
represent objects only as they appear, they are chiefly of use to 
artists ; in other words, to those who make pictures for our 
pleasure. But as projection dravnngs represent the forms of 
objects as they really are, they are more useful to industrial de- 
signers, in making patterns or copies, from which things to 
use are to be made. 

Thus, of the two figures of a goblet, shown in PI. VIII., Figs. 7 
and 8, the first represents it as it might appear standing before 
one at the table, and is a perspective view. The second is a 
projection drawing of the same goblet, showing its height, and 
its diameters at several points on its centre line or axis, on a 
uniform scale of one third of the full size. The former is 
therefore merely pictorial, while the latter might guide a work- 
man in making goblets of the particular pattern shown. 

The final drawing to be strictly followed by the workman 
might be very accurately laid out by scale, of the full size of 
the goblet and drawn with instruments. But the preliminary 
drawings, to indicate the design and its effect, would be drawn 
by the free hand, whether in perspective, or projection, or both. 
Hence, as already explained in connection with other applica- 
tions, the usefulness of the free-hand drawing of geometrical, 
as well as of natural objects. 

Indicated Exercises. Properties and Treatment of Wood. 

The following are some of the articles in the drawing of 
which the pupil can usefully exercise himself, and which each 
can generally find at hand in his home without the necessity of 
collecting models. 

First, household wares. 

Knives, forks, spoons, castors, tea sets, tubs, pumps, stoves, 
pitchers, bowls, cups, saucers, dishes, etc. 

Second, furniture. 

Under this head a qualifying remark is necessary. Wood 
may be treated in two radically different ways ; first, indepen- 
dently of its nature as having generally, or, except in root, 



64 



FREE-HAND GEOMETRICAL DRAWING. 



knot, or crotch pieces, an essentially straight grain ; second^ 
with strict reference to its structure in this respect. These 
different ways give rise to two corresponding parties relative 
to ornamental design in wood. The one, treating it as if it 
were plastic, or like marble, without a fibrous grain, pro- 
duces the kind of work often seen, abounding in curved out- 
lines and carving, as seen in curved and carved table and piano 
legs (see the annexed figure), sofa and picture frames. 




The other party claim that wood should be treated strictly 
according to its nature, that is in straight pieces running with 
the grain. Thus purely angular, not curved, geometrical work 
is produced, as seen very simply in the straight backed chairs 
of our grandfathers, and more elaborately in what is popularly 
known at present as the " Eastlake style." 

So far, however, as some kinds of wood, as ash, and others, 
are capable, at least under certain treatment, of being very 



PERSPECTIVE AND PROJECTION FREE-HAND DRAWING. 



G5 



much and variously bent without breaking, as in the " Austrian 
bent wood " furniture, shown at the Centennial Exhibition, 
and in many of our light office arm-chairs, it may be fairly 
claimed that the design does no violence to the nature of the 
wood. Also, so far as woods can be found of so close and tena- 
cious a grain, that the cutting of successive layers of grain, as 
in a tapering turned table leg, never has the effect of causing 
a splintering or peeling up of these layers where thus cut off, 




the wood seems to indicate, by its behavior, that no violence is 
done to its natural properties by curving, moulding, or carving 
it. From this point of view, the rigid exclusion of all but 
straight outlines and angularly geometrical forms from orna- 
mental wood-work, would seem to be more of a fancy, than 
well-founded in principle. 

With these explanations, the pupil will find frequent and use- 
ful examples for free-hand drawing of geometrical objects in 
articles of furniture having a regular geometrical form, such as 
Chests — Clocks — Work-boxes — Book-cases — Tables — Desks— 
Etc. 



CHAPTER III. 

PICTORIAL PROJECTION SKETCHING. 

Definitions and Principles, 

There are special kinds of projection drawing (Chap. II. n.) 
which combine the exactness of representation of projection 
drawing, with a good measure of the pictorial effect of per- 
spective drawings (Chap. II. i.), especially as applied to small 
objects. They are therefore highly appropriate for the free- 
hand sketching of such objects. 

Without going into the principles and details of this subject 
here, enough will be explained by illustration, to enable the 
pupil, beginning by imitation of copies, to serve himself suffi- 
ciently until he studies the subject fully.* 

In PL VIIL, Figs. 9 to 14, illustrate these pictorial projec- 
tions by the most elementary examples. 

The two first, or Figs. 9 and 10, represent a cube already 
otherwise shown in PL YIIL, Figs. 1 and 2. 

Isornetrical Drawing. 

Here the first figure represents a solid cube, placed so that 
the three plane right angles, a, b, c, which unite to form the 
solid angle of the cube nearest the eye, are equally exposed, 
and hence appear of equal size. Hence this kind of drawing 
is called isornetrical drawing, the name meaning equal mea- 
sure. The second figure represents a skeleton cube of the 
same size as before. Owing to the entirely regular form of 
the cube, the equality of all its sides and angles, the most re- 
mote corner, at which the three edges, de,fg, hk, meet, appears 
to coincide with the nearest one, abc. 

* See Isornetrical Drawing, etc., in my "Elementary Projection 
Drawing." 



PICTORIAL PROJECTION SKETCHING. 67 

Ex. 133. Draw the cubes as here shown, but much larger. 
The angles at a, b, o, in the drawing, are equal, and of 120° 
each. The other angles are of 60°, or 120°, as shown. 

Oblique Projection. — Again, instead of supposing the object 
to be directly in front of the eye, but inclined as just explained 
and illustrated, we may suppose the object to be above or be- 
low the eye, and also at the right or left of it. The object is 
also still supposed to be at so great a distance from the eye that 
its parallel lines will appear parallel. The four figures, 11 to 
14, of PL VIII., illustrate this case, by the representation of a 
half cube seen in as many different directions. They are called 
oblique projections to distinguish them from the preceding, and 
"because the supposed position of the body causes it to be 
viewed obliquely. 

All lines in the isometrical figures, in other directions than 
those shown, would appear less than their real size, as in case 
of one from a to e, or greater, as one from d to g. The like is 
true of all lines in the oblique projections, except all lines in the 
front faces A, B, C, D ; which show their real form as well as 
dimensions. 

Ex. 134. Fig. 11, represents the half cube as seen from be- 
low and to the riffht. Draw also the whole cube. 

Ex. 135. Fig. 12, the same as seen from below and to the 
left. Draw likewise any other square prism. 

Ex. 136. Fig. 13, as seen from above and to the right of 
it. Draw also the whole cube. 

Ex. 137. Fig. 14, as seen from above and to the left of it. 
In each case the line ac, on the body itself, is perpendicular to 
the front face of the body, and on the drawing, is made equal 
to i ab. 

The edges which are parallel to ac on the block itself, are 
parallel to ac in the drawing, on account of the supposed very 
great distance of the eye from the object, although they would 
appear to converge in a true perspective drawing (Chap. II. i.). 
Nevertheless, these figures, as well as the isometric ones, have a 
pictorial effect which makes them more intelligible to those 
workmen and others who have little familiarity with drawing, 



68 FREE-HAND GEOMETRICAL DRAWING. 

than ordinary projections are (Chap. II. n.); while they possess 
the practical advantage of showing the three dimensions of the 
object in their real size. 



Practical Applications. 

Suppose, now, that a person wishes to have made for him, a 
covered rectangular tank, with a raised square covered opening 
near one end. The figures, 1-4 of PL IX., afford a connected 
illustration of the kinds of drawing already explained. 

Ex. 138. Fig. 1, is & perspective view of the tank, not show- 
ing any of its dimensions truly. To appear not distorted, the 
eye should be about four inches directly in front of a point a lit- 
tle above E. 

Ex. 139. Fig. 2, shows a pair of projections, together rep- 
resenting all the dimensions on a uniform scale ; the lower 
one, A, is a top view, called a plan, and shows the length and 
width of the tank ; the upper one, B, is a front view, called the 
elevation, and shows the length and height. 

Ex. 140. Fig. 3, is an isoinetrical figure, where all the angles 
are right angles on the object, but appear as angles of 60° or 
120° in the drawing. 

Ex. 141. Fig. 4, is an oblique projection of the same tank. 
Both of the last two figures have nearly as much of intelligible 
pictorial character as the first figure, with the added practical 
advantage of showing all the dimensions as truly as the projec- 
tions in the second figure do, and more intelligibly to ordinary 
eyes. 

The remaining figures of PI. IX., show a variety of applica- 
tions of the projections now explained. 

Ex. 142. Fig. 5, shows a nut A, and bolt B. The nut is 
six-sided, and, according to the properties of a prism of six 
equal sides, the lateral faces, C and D, each appear just half as 
wide as the middle one. By experiment with a regular six- 
sided prism, the pnpil can easily find how the nnt would ap- 
pear if only two of its faces were visible. 

Ex. 143. Fig. 6, shows a partial plan, B, and elevation, A, 
of a square nut, the plan or top view being made first, its 



PICTORIAL PROJECTION SKETCHING. 09 

corners being, as shown by the dotted lines, a guide to the posi- 
tion of the vertical edges of the elevation. 

Figs. 7 and 8 show how to make correct isometrical drawings 
of objects whose lines are not all at right angles to each other. 
These lines, as indicated by the faint lines and the letters, are 
located from lines parallel to those in Fig. 3, since only such 
lines show their real size. 

Ex. 144. Fig. 7, represents a frustum of a square pyramid ; 
drawn, as indicated, by inscribing it in a prism of the same 
base and altitude. 

Ex. 145. Fig. 8, a fire-place. Draw Fig. 8 first as shown, 
and then with the longer edges, as MN, in the direction 
MO, which will bring the opposite end of the fire-place in 
sight. 

Figs. 9, 10, 12, illustrate the isometric drawing of curved 
objects. Each face of a cube, Fig. 9, being a square, a circle 
can be inscribed in it, touching each side of the square at its 
middle point. 

Ex. 146. Fig. 9 shows the appearance of the isometrical 
drawing of a cube with circles thus inscribed in its three visi- 
ble faces. Note that the diameter of the circle is equal to the 
side of the circumscribing square. 

Ex. 147. Fig. 10 shows a frustum of a cone, isometrically, 
its bases being drawn by means of their circumscribing squares 
which are placed in the same position as abed in Fig. 9. The 
axis, MN, joining the centres, M and N, of the bases of the 
frustum, might have been in the direction of cb, Fig. 9. Then 
the circumscribing squares of the bases would have been in the 
position cdef, Fig. 9. MN" could also have been in the direc- 
tion of cd, Fig. 9, and these squares would then have had a 
position like bceg in that figure. The pupil should draw the 
frustum in the three positions here described. 

Ex. 148. Fig. 12 is the isometrical drawing of a pipe with a 
flange at each end, the axis of the pipe being in the direction 
of cd, Fig. 9. According to the explanations just made, draw 
this pipe as it would appear with its axis lying in the direction 
cb or ce, Fig. 9. 

Ex. 149. Finally, Fig. 11, is an oblique projection of a pipe 



70 FREE-HAND GEOMETRICAL DRAWING. 

similar to the last. The figure is more easily drawn, since the 
circular ends are circular in the figure. 

Ex. 150-153. Draw Figs. 7, 8, 9, and 10 in oblique projection. 

By careful comparison of Figs. 5 to 12 with each other, and 
with 2, 3, 4 as standards, the pupil can learn to make plan and 
elevation, isometrical, and oblique projection sketches of many 
common objects; such as boxes with compartments, etc., etc. 

At this point, the pupil can also profitably, with reference to 
the exercises to follow, practice the drawing of various curved 
objects in isometrical and oblique projection. Such objects 
may be a Cylinder, a Cone, the Frustum of a cone, Rings, both 
of square and circular cross-section, Yases, etc. These may be 
taken, first, separately, and then in combination ; placing them 
first in the simplest positions, and thence advancing to more 
irregular positions.* 

The drawings in these cases should be on plates twice as 
large as those of this book, and the models should be large, 
from ten to sixteen inches high. 

Figs. 3 and 5, PI. X., are examples of the application of 
oblique projection to the drawing of models of problems in 
space, either on the black-board, or for book illustration. 

Ex. 154. Fig. 3 represents two planes, H, horizontal, and V, 
vertical, so placed that V is in the position of the front faces 
of the blocks shown in PI. VI1L, Figs. 11-14. ABC is then 
a square cornered block, whose front face is parallel to Y, and 
top, ABD, parallel to H. 

The figure abd, equal to ABD and directly under it, on H, 
Is then called the " horizontal projection," or "jplan " of the 
block. 

Likewise the figure a'b'e', equal to ABC, and directly be- 
hind it, on Y, is called the "vertical projection," or "elevation" 
of the block. See PL IX., Fig. 2. 

Ex. 155. PL X., Fig. 5, represents similar planes, IT, and Y, 
in the positions parallel to the right hand and top faces of the 
blocks, shown in Figs. 11 to 14 of PL IX., and viewed as in- 
dicated in Fig. 13. Then PQP' represents a plane, as of 
paper or tin, meeting H in the line (" trace ") PQ, and meet- 

* See note on p. 58. 



PICTORIAL PROJECTION SKETCHING. 71 

Ing V in the line P'Q. This plane is then pierced at C (whose 
projections on II and V are c and a') by the line A 13, whose 
projections are ab and a'b' . 

Figures like these are highly useful in representing combi- 
nations of forms in space, such as would not be so readily 
intelligible from ordinary diagrams. Most of the figures in 
" solid geometry " can advantageously be drawn in this way. 

The remaining figures of PI. X. give examples, but of de- 
tails, only, and on a suitable scale for practice, of the structure 
and machine sketching, which is one of the several practical 
applications contemplated in the more elementary drill afforded 
in the earlier chapters of Part I. 

Fig. 1, as indicated by the braces, shows three views of the 
assemblage of parts at one of the joints in the floor of a kind 
of bridge. 

Ex. 156. The lower figure is a sketch, as seen in a side view 
of a bridge, of the structure of the floor at the point where one 
of the main cross-beams is suspended by iron rods, as R, from 
the supporting frame, or truss above. This beam is composed 
of the four pieces whose ends are shown. The lateral ones, A, 
13, bolted to the deeper central ones, form rests for the floor- 
joists, m, which lie lengthwise of the bridge and support the 
floor planksjp. The irregular cast iron block below, is shaped 
with wings (3" x 8") which support the iron links LL, which 
tie the bridge together like a bow-string ; and form a bearing 
below for the nut on the lower end of the suspension rod R. 
Directly over this, the " plan " shows only the chain links L 
and the under side of the cast iron block, C. 

At the right of the plan is shown the view of the several 
parts as seen in looking at the end of the bridge. This figure 
should be viewed, looking to the right, while facing the left 
hand end of the plate. The same parts bear the same letters 
in the three views embraced in the figure. 

Views of different sides of the same thing should be drawn 
on the same scale, and placed on the same level, when not ar- 
ranged as in Fig. 1. The eye can then pass readily from one 
view to the other, and trace the corresponding views of the 
same parts. 

Ex. 157. PI. X., Fig. 2 shows a method of firmly binding 



72 FREE-HAND GEOMETRICAL PEA WING. 

together two timbers, M and N, at right angles, without cut- 
ting either of them. This is here done by harpoon bolts, one 
of which is A, A', where A' is a view at right angles to that 
shown at A, and shows that the upper end of the bolt is beaten 
out flat. The bolts pass through N, the hook hti gives them a 
hold on M, which is secured by the small cross bolt, b. 

Ex. 158. PI. X., Fig. 4 shows a plan, or top view of a joint 
of the iron truss of the same bridge to which Fig. 1 belongs, 
and at the upper end of one of the rods, as R, Fig. 1. 

Ex. 159. Fig. 6 shows the stout casting, called a shoe, which 
receives the ends of the last pair of tie rods, T, lying in the 
direction of LL, Fig. 1 ; also the foot of the iron truss. 

Ex. 160. Fig. 7 is a plan and elevation of a " shaft coup- 
ling," or, one form of the many contrivances for locking to- 
gether two pieces, A and B, of a line of shafting. In this 
example, this is done by means of a stout collar, C, in two 
pieces bolted together, and a key, k, partly let into the shafts 
and partly into the collar. 

On the preceding figures, the measurements indicate by 
arrow-heads the distances or dimensions to which they refer, 
and denote feet by single accents, and inches by two accents. 
Thus, 3' : 6" is read, three feet and six inches. 

Guided by these examples, the learner should exercise him- 
self in making neat pencil sketches, after the manner of those 
on Plates IX. and X., but larger, of such mechanical objects 
as are accessible, such as railroad chairs, frogs, and switches ; 
grindstones, hay-cutters, bridge joints ; roof -framings, as found 
in barns, attics, etc. 

These sketches should be large enough to exhibit the small- 
est parts, and contain the recorded measurements without con- 
fusion, or obscurity. 

By practice in thus carefully sketching, and neatly and com- 
pletely measuring the parts of any structures accessible to him, 
of wood or iron, or masonry work, or machines, or parts of 
them, the learner will not only learn to make such sketches 
readily and neatly, which will often be a serviceable accom- 
plishment, but will, by degrees, collect an album of valuable 
examples of construction, the exact knowledge of which mav 
be useful. 



r 









PL/vin. 




PL.X. 




PART III. 

ELEMENTS OF GEOMETRIC BEAUTY. 



CHAPTER I. 

ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 

1. The human mind everywhere contains among its posses- 
sions, the idea of beauty. 

Thus, it is familiar to every one that there are a multitude of 
objects which give us great pleasure, independently of any 
practical use that they have. That quality of objects, by rea- 
son of which they afford this pleasure, is called their beauty. 

2. It may not be possible yet to give a definition of beauty 
that shall include every possible case. But it seems highly 
probable that it consists in expressiveness of good, where goo© 
is defined as perfection of structure, or being • of action, or 
doing • and of consequent acquisition, possession, or having. 

3. Examples. — The beauty of a goblet, as distinguished from 
a tin cup — though the latter, being more durable and capable 
of many more uses, has far more utility — lies in part in its ex- 
pressiveness of the fitness of things, which is a kind of good. 
That is, water being transparent and colorless, yet sparkling, 
there is beauty in the idea of drinking it from a material like 
itself in these respects, from a vessel which seems as if made 
of water that had become permanently solid. 

Again, the beauty of a statue of perfect youth, so life-like that 
it makes even marble seem flexible, consists in its clear expres- 
siveness of the suppleness, readiness, abounding life, and varied 
capability which are so many perfections of being. Also, in 
its expressiveness of the mastery of mind over matter, shown in 
the genius of the sculptor. 
4 



74: FREE-HAND GEOMETRICAL DRAWING. 

Once more, the beauty of a greyhound consists in the evident 
expressiveness, in every line and conformation of his body, of 
the word, " go," whispered in his ear at his creation, as seen in 
the evident delight with which he runs a race with every fleet 
horse that passes his gate. 

4. But there are many orders of beauty, corresponding to as 
many departments of thought. One of these relates to beauty 
of form. 

Beauty of form is of two kinds; the one free, flexible, 
mobile ; the other rigid, precise, fixed. The beauty of a 
flower, or a statue, is of the former kind ; that of a building or 
a pavement of geometrical inlaid work, is of the latter kind. 

A statue is placed under the first head, not because it is 
flexible, but because it represents flexible beauty, and, in pro- 
portion to its perfection, represents it so perfectly as to seem 
flexible; so that the climax of the sculptor's art lies in such 
completeness of triumph of mind over matter as to make rigid 
material seem flexible. 

5. Of these two species of beauty, one, as we have said, 
relates to regular, or as it may therefore be called, geometric 
beauty. 

This alone concerns us now, as appropriately connected with 
a course of free-hand geometrical drawing, auxiliary to a course 
of instrumental geometrical drawing. 

Now the exactness of everything geometrical renders it cer- 
tain that, if definite principles and resulting rules can be found 
anywhere, by following which beautiful forms can certainly be 
produced, it will be in the department of geometric beauty. 

There is reason to suppose that the ancient Greeks possessed 
and used such rules, unless their marvellous genius for such 
beauty, as well as for purely free or non-geometrical beauty, 
made them infallibly, though unconsciously, conform to them. 

6. Let us next examine this idea of beauty and seek to find 
some of its parts or elements. 

Unity. — Passing by a large vacant city lot at a certain time, 
we may notice in process of collection a large amount of scat- 
tered stone — iron — lumber — lime. 

Passing the same spot a year or two later, we .find that aL 7 



ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 75 

these materials, with other finer ones, have been combined to 
form a grand temple of art, science, or religion. 

The human mind, of itself, feels and knows a difference 
between these two cases, and expresses the difference in appro- 
priate speech. 

The first is an assemblage of materials for a purpose not yet 
realized. This assemblage is, to the mind, no one thing, and 
takes no distinctive name as any one thing. 

The finished building embraces the same materials, but com- 
bined for one purpose. This one purpose, governs the orderly 
arrangement of the before scattered materials, and makes each 
piece contribute in some way towards the fulfilment of that 
purpose. This fact makes the result one thing to the mind, 
to which one name can therefore be given — a museum, or a 
church. 

Hence we say, that unity is one of the primary ideas of the 
human mind, and that a principle of unity pervades and binds 
together things otherwise thought of as separate. 

7. Illustrations. — It is this same idea which makes the dif 
ference between anarchy and civil order; between chaos and 
creation, and which makes the mind always conceive of the 
sum of all things as being really one thing, because the product 
of one mind, for some one all-embracing purpose, and hence 
called the universe. 

Finally, there is no stronger proof of the permanent reality 
of this idea of unity, than the existence, always and everywhere, 
of systems of philosophy. For philosophy, as is plain from 
many definitions of it, is an attempt to discover the central or 
initial thought, purpose, or idea, from which all things visible 
and invisible spring. In other words, it is an attempt to see 
the universe as from its centre ; in a word, to place ourselves 
in the position of Deity, with the intelligence of Deity — as 
Plato said 2,000 years ago, " a resembling of the Deity so far 
as that is possible to man." 

8. Kinds of Unity. Uniformity. — But unity is of two 
kinds, simjyle and compound. To continue a former illustra- 
tion, one brick, one board, in the building is a simple unit, as 
being of uniform substance throughout, and not formed by put- 
ting together separate pieces. On the other hand, the entire 
building is a unit, because made for one purpose, and evidently 



76 FREE-HAND GEOMETRICAL DRAWING. 

made for one purpose, in that every part of it contributes to* 
wards the attainment of that one purpose. Yet it is a highly 
compound unit, because composed of many separate parts. 

When the separate parts are like the whole, or when, if the 
nature of the case renders this impossible, the like components 
are equal and similarly placed, the principle of uniformity 
enters, and the unity, though still compound, is simplified, or 
more nearly approaches simple unity. 

Hence we have not only unity, but unity in variety , with, or 
without uniformity also, as an element in the idea of beauty. 
As each part co-operates with the others to form the unity of 
the whole, we will call this compound unity, harmony. 

9. Freedom. — But there is variety in a higher sense. There 
may be twenty great buildings all built for the same purpose. 
Yet they may all be well adapted to that purpose and hence 
beautiful, though built in twenty different ways, or from the 
plans of twenty different designers, none of whom ever saw 
the plans of any of the others. 

This kind of variety may be a part of what is meant when 
writers on Art speak of the " freedom of the domain of art " 
as compared with the precise rules by which we are bound in 
mathematical operations ; and when they speak of the freedom 
of the mind over matter, when the mind seeks to express its 
ideas and purposes by material forms. 

This variety indicates the principle of freedom, since it re- 
sults from such action as is most unconscious of conformity 
to rules, and is most difficult, if not impossible, to bring under 
the operation of rules. 

10. Primary relation of fundamental proportions to acces- 
sories. — In analyzing the variety just described, nothing is more 
familiar than the habit of distinguishing between the propor- 
tions and the decorations of a structure. And no principle is, 
or should be, more familiar than that no kind or amount of de- 
coration can conceal or compensate for deformity of the naked 
proportions, skeleton, or framework, which supports those de- 
corations. 

Now the very idea of proportions, is, that they are something 
of a definite geometrical character, having precisely measura- 
ble, or numerical relations between them. This being true, it 
follows that the skeleton — that is, the figure composed of the 



ELEMENTARY IDEAS. UNITY, VARIETY, FREEDOM. 77 

principal lines of any structure, or other object, should form a 
geometrical figure governed by the laws of agreeable geometri- 
cal proportion ; supposing, as in Art. 4, that it is possible to 
discover these laws. 

11. Summary. — We are now prepared to enter upon the 
study of geometrical beauty, guided by the three principles of 
Unity ; Harmony ; and Freedom, such as maybe exercised 
upon an underlying frame or' skeleton having beautiful geome- 
trical proportions. 

These terms, harmony and freedom, are not inconsistent, as 
should here be understood, with the terms, symmetry and com- 
bination, before used (see Part I., Chap. VII.) ; for symmetry 
is but one species of harmony, and freedom consists partly in 
the multitude of combinations which can be made from the 
same elements. 



CHAPTEE II. 

NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. 

12. Yitruvtus is the one ancient writer on architecture and 
its details, to whom modern writers refer ; and his knowledge 
of the principles of Greek art seems to have been traditionary 
and incomplete. Yet he says, u The several parts which con- 
stitute a temple ought to be subject to the laws of symmetry ; 
the principles of which ought to be familiar to all who profess 
the science of architecture. Proportion is the commensura- 
tion of the various constituent parts with the whole, in which 
symmetry consists."- And he then describes the details of the 
proportions of the human body, as being the most beautiful 
created thing ; and says " the laws of symmetry were derived by 
the artists of antiquity " (the Greeks of whom he wrote) " from 
the proportions of the human body." 

13. But how these laws of symmetry were derived from the 
human body he does not show. 

Nevertheless we have the following clue to their possible 
discovery. 

1°. The eye and the ear are sometimes spoken of as the 
nobler senses, the especial senses of the soul. 

2°. There are accordingly provided, to satisfy these senses, 
beauty of sound for the ear, and beauty of form for the eye. 

3°. The laws of concordant sounds, harmonious to the ear, 
are well known. If then, taking the human body as an illus- 
tration of the utmost beauty of form, its proportions should be 
found subject to the same laws as those of harmonious sounds, 
the principles of unity and harmony already explained would 
make it seem highly probable, that the true principles of exact 
or geometric beauty olform were the same as those of exact or 
harmonic beauty of sound. For, moreover, the principle of 
freedom, as well as of mathematical precision, enters the do- 
main of both eye and ear. This is seen in the beauty of oratory 



NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. '(7t 

and elocution, as distinguished from music, properly so called, 
which corresponds, it may be, with the beauty of natural ob- 
jects as distinguished from geometrical ones ; or more exactly, 
in the fact that various pieces of music may yet be, in some 
evident manner, appropriate to the same words. 

14. Direction, the primary element of Form. — The two 
elements of form are direction and length. The following con- 
siderations seem to show that direction is the more fundamental 
of these. 

First, practical considerations. If the inquiries of a hun- 
dred travellers seeking their way to an unknown location were 
noted, it would probably be found that they would ask " which 
way," before they asked " how far." 

Again in describing a survey, it has long been customary to 
describe the direction, called the bearing, of each line, before 
stating its length. 

Once more, in making any ornamental design, whether of 
regular or free outline — but especially in the latter case, as in 
sketching fruit and flower forms or scroll work — the mind is 
more occupied with the direction to be given to the pencil at 
each point of its progress, than to the size of the sketch to be 
produced. 

Second. But a precise geometrical reason may be found in 
the fact, that the triangle, the fundamental figure into which 
all others may be decomposed, may have an infinity of sizes, 
all of one shape, but cannot have any variety of form with a 
fixed length for each side. 

That is, in similar triangles, similarly placed, the correspond- 
ing sides may be of different length, but have the same direc- 
tion. 

15. It may be objected that there can be many triangles of 
the same size or area, but of different forms, as well as many of 
the same forms but of different sizes, and thus that the ideas 
of direction of sides, and length of sides are equally funda- 
mental. But it is to be noticed that the similar forms of dif- 
ferent sizes can be instantly perceived to be similar, while the 
equal sizes, of different forms, could only be known to be equal, 
by measurement and computation of their areas. 

Also, if one were trying to sketch a symmetrical, that is an 



80 



FREE-PIAND GEOMETRICAL DRAWING. 



isosceles triangle of pleasing form, he would do it by trying 
various angles between the base and the adjacent sides, as in- 
dicated at A, Fig. 1, until he found a triangle of pleasing 
proportions ; rather than by trying various given lengths, as 
c and d, of those sides, placing them together by means of 
dividers, as indicated at BC and BD. 




A B 

16. Summary. — Distance and Direction, are two radical 
geometrical ideas. Distance lies at the foundation of size, as 
large or small. Direction lies at the foundation of form, or 
the shape of things ; and it is the form of objects rather than 
their size, which determines their beauty. Hence direction 
appears as the root idea in geometrical forms. 

17. An angle is difference of direction, or the measure of 
relative direction. Hence it seems natural to look for the 
principles of geometric beauty in the ratios between the angles 
of figures, rather than in those between the lengths of their 
sides. 

Some remarkable numerical properties of the circle, as the 
measure of angles at its centre, will confirm this view. 

18. The circle has for ages been divided into 360 equal 
parts, called degrees, for the purposes of angular measurement. 
Whether the selection of this number was the result of acci- 
dent, or experiment, or of abstract reasoning, may not now be 
known ; but its relation to the principles of unity, variety, and 
harmony, as expressed by numbers, is very striking. 

First. Its prime factors are 

1. 2. 2. 2. 3. 3. 5 = 360. Now, 
1°. These factors are the abstract unit, 1, and its first three 
prime multiples. 



NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. 81 

2°. Of those factors, 2, the first even number, consists of the 
two equal halves 1 and 1. It is therefore expressive of the 
principle of uniformity, as in the division of a body into two 
equal halves. 

3°. Next, 3, is the first odd number. It can be separated only 
into the equal parts 1, 1, 1, or the unequal ones 1 and 2. It is 
thus the first and simplest numerical representative of the 
principle of variety. 

4°. Next, 5, is the second simple or prime multiple of 1, and 
the first and simplest which combines in itself the numerical 
representatives of uniformity and variety, 2, and 3. It is there- 
fore the simplest numerical representative of the combination 
of uniformity with variety. 

19. The factor 2 is the foundation of the series 2, 4, 6, 8, etc., 
made by taking 1, 2, 3, etc., successively as multipliers. The 
factor 3 is likewise the foundation of the series 3, 6, 9, 12, etc.; 
and 5, is likewise the first term of the series 5, 10, 15, etc. 

The numbers afforded by these series can, however, be better 
exhibited with regard to the details of their dependence on the 
primary numbers 2, 3, 5, as follows, where the exponent of 
each primary number is the multiplier for the series begun by 
that number. 

2 a , 4, 8, 16, etc. 

2 3 , 6, 18, etc. 

2 5 , 10, 25, etc. 

3 2 , 6, 12, etc. 

3\ 9, 27, etc. 

3 6 , 15, 75, etc. 

5 2 , 10, 20, etc. 

5 3 , 15, 45, etc. 
5 6 , 25, 125, etc. 

20. The factors of 360, properly combined by multiplication, 
yield all of the nine digits except 7, viz., 1; 2; 3 ; 2x2 = 4; 
5; 2x3 = 6; 2x2x2 = 8; 3x3 = 9. 

The same numbers are also found in the foregoing series. 
But seven is peculiar, in containing none of the radical num- 
bers 2, 3, 5, as a factor, and as a sum, 3 + 2 + 2, it is redundant 
as compared with 5, in containing 2, twice. To be sure, 8 as a 
sum, 2 + 3 + 3 is redundant, though differently, but can be re- 
solved into 2's, it being 2 x 2 x 2, as 9 can into 3's, while 7 is 

impracticable. 
4* 



82 FREE-HAND GEOMETRICAL DRAWING. 

"We conclude therefore that 2, 3, 5, are the primary numeri- 
cal representatives of uniformity and variety, and of their com- 
bination without superfluity ; and thence that 360 may have 
been chosen to express the divisions of a circle on account of its 
containing all the digits but 7.* 

21. It is interesting to note in passing, that the remaining 
digit, 6, is the first perfect number, that is, one which is equal 
both to the sum, and to the product, 1 + 2 + 3, and 1 x 2 x 3, of 
its factors, and thus numerically represents, both as a sum and 
a product, the principles of unity, uniformity and variety and 
their combination. 

22. Having thus arrived at the significance of the numbers 
2, 3 and 5, the number of ways and the manner in which 360 
contains them is quite surprising, as is distinctly shown to the 
eye, in the annexed table. 



360=2x2x2x45. 


(1) 






360=3x3... x40. 


(2) 


-^T"^* 


360=5 x72. 


(3) 


s'~ 


*^v 


72=2x2x2x9. 


(3) 


it 


\ 


45=3x3... x5. 


(1) 


t 

i 


\ 
I 


40=5 x8. 


(2) 


\ /^\. / 


8 = 2x2x2x1. 


(2) 


3\ /* 


9=3x3... xl. 


(3) 


n^ y 


5 = 5 xl. 


(1) 







That is : First, 360 contains, as before seen, 2 thrice as a fac- 
tor; 3, twice and 5 once, with quotients, after successively di- 
viding out all these factors, of 45, 40, and 72, in the three cases 
respectively. Second, these quotients contain the same factors 
in like manner, as shown in the second group. Third, the like 
is true again, taking the quotients of the second group in the 
order seen in the third group, where the final quotients are 
each, 1. 

In this curious result, we see again, exhibited in numbers, the 
principle of uniformity. 

But the order in which the first two groups of quotients are 
used as the next dividends, is found as indicated by the paren- 

* The number 7 is also excluded generally in the formation of musical 
ratios, but is said to be employed in Chinese music, and it enters into the 
composition of some peculiar theoretical systems, not in actual use. 



NUMERICAL EXPRESSION OF THE ELEMENTARY IDEAS. 83 

theses and the adjoining circle, by combining 1, 2, 3 in every 
possible order taken in rotation, in the direction of the .arrow. 
Here again we have likewise the principle of variety. 

23. Ratio the principle of combination. — Numbers can be 
compared in two ways, by difference and by ratio ; difference 
being obtained by subtracting one number from another; and 
ratio, by dividing one by the other. Aristotle (as quoted by 
Hay) defines harmony as " the union of contrary principles " 
(as those of uniformity and variety) "having a ratio to each 
other." In beautiful forms, proportions constitute harmony; 
and Yitruvius defines harmony as " the commensuration of the 
various constituent parts with the whole ; " that is, each part 
bears a certain ratio to the whole. 

21. The reason why ratio, rather than difference, should be 
the combining principle of parts into a whole, seems to be that 
difference belongs to the domain of things, and ratio, to that 
of pure thought. That is, this is so in this sense, that things 
of a kind can be added or subtracted forming a greater or less 
number of things of the same kind, as 5 pounds + 3 pounds 
are 8 pounds ; 10 feet — 2 feet are 8 feet, etc., processes which 
seem to imply after-thought, or the putting together, or put- 
ting apart of things complete in themselves. 

But, on the other hand, we cannot multiply 5 pounds by 3 
pounds, or divide 10 feet by 2 feet, and get any real result. 
We can however multiply 5 pounds by the abstract number, 
3, giving 15 pounds, and divide 10 feet by the abstract num- 
ber, 2, giving 5 feet. 

Addition thus seems related to the miscellaneous assem- 
bling of the materials supposed in Art. 6, but ratio to their 
combination, primarily in the mind, and then realized by the 
hands, in a thought-out system, which is a unity, and not an 
assemblage ; though a compound unit, in which however each 
component contributes in its proper degree to the intended use 
of the whole, as when the windows of a building are sufficient 
for its light, and its entrance doors sufficient for ingress and 
egress. 



CHAPTER III. 

GENERAL APPLICATIONS OF THE IDEA OF BEAUTY IN RATIOS. 

Analogy of Form and Sound. 

25. Lineal" and superficial beauty. Since ratios, rather 
than differences, determine harmonious proportions, beauty 
arising from marked divisions of a line, will consist in ratios 
between the parts of the line, or between the parts and the 
whole ; ratios which, according to the last chapter, are derived 
from the numbers 1, 2, 3 and 5. Such ratios will be furnished 
to any desired extent by the several series given in Art. 19. 

Again, coupling the principle of ratios, with that of direc- 
tion, as more fundamental than distance, the ratios between 
the angles of a plane figure would determine the beauty of its 
proportions, rather than the ratios between the lengths of its 
sides. It only remains then to determine the natural unit of 
angular measure. 

26. The angular unit. This unit must be simply an angle 
that is naturally, not arbitrarily fixed. Recurring then to the 
figures in Part L, Chapter II., p. 7, we see that when hg is 
moved either way from its position perpendicular to mk, the 
angle on one side of it is acute, and that on the other, obtuse. 
Acute angles may vary in size, infinitely between 0° and 90°, 
and obtuse angles may vary indefinitely between 90° and 180°. 
But the equal angles of 90° on each side of hg, when it is per- 
pendicular to mk, are the limit between all acute and all 
obtuse angles. That is, the right angle, being of necessarily 
fixed size, is the standard of comparison for all other angles. 

27. Beginning now with known principles, from which to 
proceed to the unknown, the first illustration of beauty sensi- 
bly derived from the divisions of a line, sir ill be the beauty of 



GENERAL APPLICATIONS OF THE IDEA OF BEAUTY IN RATIOS. 85 

sound,' occasioned by the notes given ont by the divisions of a 
vibrating string. 

By showing that the long and well established laws of musi- 
cal harmony, or beauty of sound, are based upon the numbers 
2, 3, 5, and their multiples, it will be more readily apparent 
that the abstract principles of beauty already described in con- 
nection with these numbers, are also as truly the bases of geo* 
metrical harmony, or beauty of geometrical form. 

That the musical terms employed may be better understood, 
a sketch of a portion of a piano or organ keyboard is here 
given. Fig 2. 



p T> E > $ fc B \> 






Ffg.a. 




a c p e r g a ja c, d, e, f, q, a 3 



28. A note occasioned by a certain number of vibrations per 
second (fixed upon by agreement), whether of a stretched 
string, as in a viol, harp, or piano, or of a column of air, as in 
an organ pipe, is adopted as a standard of comparison, and 
designated by the letter C A string one-half as long, Fig. 3, 
vibrates twice as fast, and gives a note designated as Q x , which 
is described as an octave above C In other words, the differ- 
ence in sound between these notes is called the interval of an 
octave. 

6 & £ 



Fig- 3 



Again : f of the length of the string will vibrate three times 
while the whole string vibrates twice, that is f times while the 
whole string vibrates once, and will yield the note designated 
as Gr, and described as being at an interval of a fifth above C. 

'29. By continuing to take different simple fractional parts 



86 FREE-HAND GEOMETRICAL DRAWING. 

of the string, based, according to preceding principles, on the 
numbers 2, 3, 5 and their multiples, we find the results given 
in the following table. 

The first column contains the order of the notes and the 
letters which designate them ; the second, the number of vibra- 
tions made in producing each note, during one of the vibrations 
made in producing the note C; the third, the number of vibra- 
tions made in producing each note, beginning with D, during 
one of the vibrations belonging to the next preceding note • the 
fourth, the name of the interval between each note and the first ; 
the fifth, the name of the interval between each note after the 
first and the next preceding, these intervals being those which 
are expressed numerically by the ratios of the second and third 
columns respectively. The ratios in the second column are ob- 
tained experimentally. Those in the third are found as follows. 
Taking for illustration G and A, if we call f — 1, what would -| 
become % Ans. f : 1 : : f : lx|= | x f =^-. 

3 

2 

I. II. III. IV V. 



1. C 1 1 1st 1st, or unison. 

2. D | | 2d Major 2d. 

3. E £ -LQ. 3d Minor 2d. 

4. F f f| 4th Diatonic semi-tone. 

5. G | -I 5th Major 2d. 

6. A | ^°- 6th Minor 2d. 

7. B V- i ?th Major 2d. 

8. Cj 2 -L| 8th Diatonic semi-tone. 

30. Illustration of further application of ratios, founded 
directly or indirectly on the primary numbers 2, 3, 5. 

The intervals from C to E, from F to A, and from G to B 
each consisting of a major and a minor 2d — and = f — are called 
major thirds. Those from E to G, from A to C, and from B 
to the next higher D, each consist of a major 2d, and a diatonic 
semi-tone, and are each = |. Thus G, |-^-E, f = f x f=f . 
These intervals are called minor thirds. Again, |-^J- = f -f-, an 
interval called a chromatic semi-tone. 

Once more, F, HA * = i * i = H aTld i^i = HM = 

160 — 80 
TBT — 8T- 



GENERAL APPLICATIONS OF THE IDEA OF BEAUTY IN RATIOS. 87 

Also, -J : 1 :: -V : ¥-5-f = ¥ x f = !?• This interval, f$, 
between the odd interval (D — F) and a minor third, or be- 
tween a major and a minor 2d (remembering that § -{- means 80 
vibrations belonging to one note, simultaneously with SI vibra- 
tions belonging to another note) is called a comma. 

31. Sha)ys and Flats, — A note so much higher than a given 
note that 25 of the vibrations which make it take place while 
24: of those which make the given note occur, is called the sharp 

of that note, and is marked Jjl. Also a note so much lower than 
a given note that 24 of the vibrations which make it occur dur- 
ing every 25 of those which make- the given note, is called the 
fiat of that note, and is marked £?. 



Thus we have, C 



<# 



25 
2¥ 



sJ 



D 



25 
2T 



25 

21 



H- 



\ 25 

15' 



e — i' 



Continuing this process for every note of the scale, we shall 
find, in place of the eight notes, C to C t inclusive, twenty-one 
notes, with an exceedingly complicated variety of intervals, 
arising from the various combinations of major and minor tones 
and the diatonic and chromatic semi-tones. 

32. Keys and their mutual adjustment. — J^o absolute note, 
that is, a note made by no fixed number of vibrations per second, 
need be taken as 1 of the musical scale. Any of the 21 tones 
just mentioned, may be taken as the initial note, called the hey 
note of a scale in which the order and value of the intervals 
shall he the same as in the scale just explained, in which C is 
made 1 of the scale. But in any such new scale, a greater or 
less number of the notes will not coincide with some of the 21 
notes of the complete scale which begins with C. Thus, a fur- 
ther and greater complication arises, out of the difference be- 
tween the two kinds of tones and semi-tones already described. 



88 FREE-HAND GEOMETRICAL DRAWING. 

A single illustration will suffice. 

33. Suppose the note G to be assumed as 1 of a new scale. 
Such a scale is said to be in the key qfG. 

C 
D 
E 
F 

G— G, 1. 

A — A', 2. a comma higher than A, to give a major second 
as there should be (Art. 29) from 1 to 2 of the 
new scale (Art. 32). 

B — B, 3. Minor second above A'. 

C L — C 1? 4. Diatonic semi-tone above B. 

T>i — D l5 5. Major 2d above C. 

Ei— E 1? 6. Minor 2d above D 

Fi— F/, 7. Major 2d above E x . 

G x — G 1? 8. Diatonic semi-tone above F/. 

It here appears that in the simple diatonic scale of eight 
tones, two new ones, a new A, called A' and f-J, or a comma, 
above the A of the C scale, and a new F as described, are 
necessary to make the intervals exactly the same in the scales 
beginning with C, and G, respectively. And it will be found 
that two additional notes will be required to perfect each new 
scale ; and more still would be required, if the semi-tones 
(sharps and flats) were all considered. But the intervals, very 
nearly f f- between Cjf and Dj2 for instance, are so small, and 
especially so very small between either and a note which would 
be the mean between them, that it is usual to abolish the dis- 
tinction between the sharps and flats of consecutive notes, thus 
for instance making CJJ and T>U identical, and also to abolish 
EC, F2, Bjf and Ch ; all which abridgment reduces the 22 
tones to 12, as shown in Fig. 2. 

But with two additional tones for each new key note, an 
organ giving perfect intervals in the simple diatonic scale of 
eight notes, in eight keys, or scales, besides the C scale, would re- 
quire 12,-1-8 x2=12-f 16 = 28 pipes for every twelve now used. 
Such an instrument, though once, at least, actually built, would 
be extremely bulky ; and in practice, the octave C to C x is di- 



GENERAL APPLICATIONS OF THE IDEA OF BEAUTY IN RATIOS. 89 

vided usually into 12 equal intervals, thus rendering all scales 
alike, by making all equally imperfect. 

34. Temperament. — The adjustment just described is called 
temperament, and may be broadly defined as the limitation of 
free conformity to an ideal standard, in obedience to con- 
straining conditions. Thus, music performed on stringed 
instruments, like violins or harps, may be in perfect harmony 
in all keys, since the length of string necessary to produce any 
desired note may be regulated by the fingers. But with instru- 
ments with fixed keys, as for example, those having a piano 
key-board, it is nearly if not quite impossible to arrange more 
than twelve keys to the octave, though attempts have been 
made, by dividing the black keys, for example, to distinguish 
between the sharps and the flats of the principal notes. That 
is, the harmony is constrained by the mechanical difficulties of 
the problem, either relative to a practicable key-board, or to 
the bulkiness already explained in the last article. 

Curious examples of temperament, as here defined, and an- 
alogous with musical temperament, will be found in connec- 
tion with geometrical beauty.* 

35. Passing now from sound to form, based on linear 
beauty, the following, a few among many of the linear propor- 
tions of the human form, remarkably coincide with the ratios 
existing between the vibrations in a given time, which produce 
harmonious sounds. 

Parts of the 
total height 

From the sole of the foot to the groin \ 

" " " " 5th or last vertebra of the loins £ 

" " " " top of the hip bone % 

" " " " breastbone , £ 

" " " " bottom of the jaw bone £ 

" » " " top " " " | 

" " " " base of the spine ~h 

From the base of the knee-pan to the top of the head f 

* It is of course not indispensable that the theory of music, even only so 
far as is here indicated, should be understood in order to render the succeeding 
principles of geometric beauty intelligible, since each rests independently on 
the abstract principles of beauty founded, as before shown, on the numbers 
2, 3, and 5. Only, if it be plainly seen, that musical harmony is expressed by 
ratios founded on these numbers, it will more readily appear that geometric 
beauty may be similarly founded. 



CHAPTER IV. 



APPLICATION TO TRIANGLES AND RECTANGLES. 

Triangles. 

36. Proceeding from linear to superficial beauty, and bear- 
ing in mind the principle of Arts. (16) and (17) we shall find 
its simplest geometrical expression in triangular figures whose 
angles have simple ratios to each other, founded on the num- 
bers 2, 3, 5. And, of these triangles, right angle ones are the 
simplest in relation to our present subject, since they contain in 
themselves the standard angle of comparison for the other two 
angles. (26) 

Formal beauty, thus founded upon the numbers 2, 3, 5, may 
be said to be of the first, second and third orders respectively. 

37. The simplest geometric beauty of the first order will be 
represented by the isosceles right-angled triangle, Fig. 4, in 




which the equal acute angles are to each other as 1 : 1, and to 
the right angle, as 1 : 2. Also this triangle can, as shown at 
A a C and a b B, be indefinitely divided into triangles similar 
to the whole one, ABC. It thus geometrically represents the 
principle of uniformity. 
. 38. The simplest geometric beauty of the second order is 



APPLICATION TO TRIANGLES AND RECTANGLES. 



01 



that of the equilateral triangle. For such a triangle, ABC, 
Fig. 5, is divided by its altitude CD into right angled triangles 
of 30°, G0°, 90°, giving the ratios £, £, f, thus illustrating, by 
its exhibition of the number 3, in connection with 2, the first, 
principle of variety. 




39. The simplest geometric beauty of the third order is 
that of the isosceles triangle having an angle of 36° and two of 
72°, Fig. 6. For its altitude, CD, divides it into triangles of 
18°, 72°, 90°, which exhibit the ratios J, £, \ , thus, by including 







c 




pi 




fvt 


£>0 




A 




p 



R3-.6. 

two ratios based on the number 5, illustrating the second prin- 
ciple of variety, the combination of uniformity with variety 
(Art. 18). It also, in the whole triangle, 36°, 72°, exhibits 
the ratio, \. 



92 



FREE-HAND GEOMETRICAL DRAWING. 



40. Derived ratios. But from the first example (Art. 37) i, 
the ratio of 45° to 90°, is equal to the product, f xf, where § 
may be the ratio of 30° to 45°, of 45° to 67° : 30' and 60° to 
90° ; and where f may be the ratio of 67° : 30 to 90°. Thus is 
derived a harmonic triangle of 22° : 30' ; 67° : 30' ; 90°, giving 
ratios of -J- ; J- ; j-. Likewise, in the second example (Art. 38), f 
is the product f xf; and -§- may represent the ratio of 60° to 
75°, while | may represent that of 75° to 90° ; of 30° to 36°, 
etc. Thence arises the triangle of 15°, 75°, 90°, giving the 
ratios \, ■$-, |. Lastly, in the third example (Art. 39), -f = j^o 
where f may represent the ratio 72° : 81° and T \ the ratio 
81° : 90°. 

41. This gives all the usual ratios in musical harmony, ex- 
cepting T 8 y and ||. Now, observing that the numerator of th( 
first of these ratios is based on the number 2, while its denomi- 
nator is based on 3 and 5, we have T 8 7 = f x -§- , the product of 
the two highest ratios afforded by the second and third exam- 
ples. This affords a triangle of 42°, 48°, 90°. 

A A 



f8o° 



f81* 



F7W» &« 



See now, also, Figs. 7 and 8, showing the right-angled half 
of acute isosceles triangles of 20° and 80°, and of 18° and 81°. 
These halves are right angled triangles of 10°, 80, 90°, and of 
9°, 81°, 90°. The first gives ratios 10 : 80, of | and 80 : 90, of f 



APPLICATION TO TRIANGLES AND RECTANGLES. 9.'} 

and the second, ratios, 9 : 81, of *- and SI : 90, of T V Now | = |f 
of ,-'„ and A, = yilx If- Here then we have the ratios -}; r ;, 
;,'. characteristic of the musical intervals respectively of a 
diatonic semi-tone, a chromatic semi-tone, and a comma. (Arts. 
29, 30, 31.) 

Rectangles. 

42. The next simplest figure to the triangle, and the most 
frequent one in architectnral and other designs, is the rectangle. 
A rectangle is resolved by its diagonal into two equal right an- 
gled triangles. Hence a rectangle is of harmonious propor- 
tions, when the ratios of the angles of one of its component 
triangles to each other are simple, like those already noticed. 

Rectangles enter more largely than any other figure into the 
design, of a great variety of objects ; b.uildings of all kinds, and 
their subdivisions, garden compartments; regular furniture; geo- 
metrical decorations; railroad cars, books and writing paper; 
ornamental boxes, carved chests, etc. 

With this suo-o-estion, and from the foreo-oino- sufficient state- 
ment of principles, the learner can exercise himself in design- 
ing manv common rectangular things, either by drawing them 
in the manner shown in PL IX., Fig. 2, or by ma-king paper 
models of them, and this he can do with more pleasure and enter- 
tainment, and profit, through the formation of new ideas of his 
own, than by merely copying any number of given patterns. 

Example 1. Draw all the rectangles of which Figs. 4 to 8 inclusive are 
the triangular halves. Also all those indicated in (40, 41) that is, those in 
which the ratios of the angles of the triangular halves are 3, \, \ and £, £, 
I; etc. 

43. Independent or detached rectangles, as doors, windows, 
panels, etc., can be designed in unfettered conformity to the 
foregoing elementary principles ; but rectangles forming the 
floor, sides and ends of a room are mutually dependent, and 
cannot always be strictly conformed to these principles. But 
the discords of form thus arising can be disposed of in two 
ways, at least. 

First. Wainscotings, platforms, cornices, or ornamental 
bands, may be so adjusted in position and width as to break up 
an undivided rectangle into subordinate ones of perfect form. 



94 FREE-HAND GEOMETRICAL DRAWING. 

This is somewhat analogous to the fact that in music the 
chord of the second, which, taken alone, is inharmonious, never- 
theless greatly enriches certain combinations of notes into 
which it enters. 

Second. By analogy with temperament (Arts. 32-34), which 
is mechanically unavoidable in keyed instruments, a noticeable 
departure from perfect form in some one of a combination of 
rectangles may be distributed among all of them, thus making 
it less conspicuous. 

44. Illustration. — Suppose the principal room in a handsome 
dwelling to be 36 feet long. Let its floor be divided by its 
diagonal into triangles of 30°, 60°, 90°. This will give the 
room a width of 20 ft., 7.7 ins., \ery nearly.* 

Next, taking the longest diagonal of the body of the room as 
its principal line, let this make an angle of 18° with the floor. 
This, the diagonal of the floor being about 41 ft., 3.4 ins., will 
give a height of 13 ft., 10 ins., very nearly, which is sufficient 
for an apartment of imposing proportions. This height makes 
with the width, a rectangle for the end of' the room, whose 
diagonal divides it into triangles of about 33°:30 / ; 56°: 30'; 
90°, which does not differ seriously from the very simply 
harmonious one of 36°, 54°, 90° ; or from one of 30°, 60°, 90° 

Again, the same height, combined with the length, 36 ft. 
gives rectangular sides composed of triangles of 21°, 69°, 90° 
which thus do not differ greatly from 22° : 30' ; 67° : 30', 90° ; or 
from 18°, 72°, 90°. 

45. It now appears that either a slight increase or decreas 
of the height — neglecting the intangible angle made by th 
longest, or space diagonal with the floor — would make both th 
side and end rectangles nearly perfect. 

Thus, if the side rectangle contains triangles of 22°: 30'; 
67° '.SO'; 90°, the end rectangle would, at the same time, be 
composed of triangles more nearly than now of 36° : 54° : 90°, 
and the height, thereby increased to about 14 ft. 9 ins., would 
add grandeur to the room. 

Otherwise, if the side rectangle were reduced to triangles of 
18°, 72°, 90°, the end rectangle would be, more nearly than now, 

* As found by careful plotting, on a large scale. 



APPLICATION TO TRIANGLES AND RECTANGLES. 



95 



composed of those of 30°; 60°; 90°, where, as in the previous 
adjustment, the ratios are simple and harmonious. The height, 
now reduced to about 11 ft. 8 ins., while rather low, vet helps 
to realize certain ideas of snug home shelter and winter com- 
fort, which are as agreeable in their way, in a cold climate, as 
the stateliness of a lofty room. 

46. To afford a complete view of harmony of form in com- 
bined rectangles, it is not enough to secure a harmonious form 
for each rectangle independently. The like angles of the dif- 
ferent rectangles should harmonize by having simple ratios to 
each other. Thus, in Fig. 9, let ABCII be a floor, with its 

H IB ,fe 




diagonal, AH, making an angle of 30° with its edge AJB, and 
ABGF the side wall folded down into the level of the floor, and 
with a diagonal making an angle of 22°: 30' with its base AB. 
Also, let ADEC be the end wall, similarly folded, and whose 
diagonal AF will make an angle of not far from 36° with its 
base. We shall then have the' ratios zzy^stL = £§- = f ; JL Sir- -' 
s= ^J = -|; and J-§- = -§-, all of which are harmonious by their 
simplicity and relation to the radical harmonic numbers 2, 3, 5. 

Ex. 2. Note the ratios between the other three angles indicated in the fig- 
ure by dots. 

Ex. 3. Determine likewise the ratios when the angles of 22° : 30' and 36° 
are made 18° and 30° respectively. 

Ex. 4. Construct a box, or a pasteboard model having the proportion ? of 
either of the rooms just described. 



CHAPTER V. 

GEOMETRIC BEAUTY OF POLYGONS. GEOMETRICAL DESIGN. 

47. First. "We have seen (Arts. 37-38) that the primary 
figures representing harmony of form are 1st, the right angled i 
2d, the equal angled, and 3d, the 36° and 72° acute angled isos- 
celes triangles. Then the corresponding primary rectangles are 
those made by placing the halves of these triangles together by 
their hypothenuses. These rectangles will be the square / the 
rectangle whose diagonal makes angles of 30° and 60° with its 
sides ; and the rectangle whose diagonal makes angles of 1 8° 
and 72° with its sides. 

Second. We have found (Arts. 40, 41) that from these pri- 
mary triangles others are derived, and from these, in turn, a cor- 
responding series of rectangles proceed. 

48. It is interesting next to note that most of the regular 
polygons, yield simple harmonic ratios, by means of their sub- 
division into equal triangles, by radii from their centres to 
their corners. 

49. Thus the equilateral triangle, Fig. 10, divides into three 





Fig* 



10, 



Ffcr 



. 41. 



triangle; of £0° and 120°, giving the ratios \ and ^. The 
square, Fig. 11, divides into triangles of 45° and 90°, giving the 



GEOMETRIC BEAUTY OF POLYGONS. GEOMETRICAL DESIGN. 97 

ratios -J- and J. The regular pentagon, Fig. 12, is composed of 
triangles of 54° and 72°, giving the ratio J. 




r,*, 14. 



'o 



Ex. 5. In like manner note the angles and ratios afforded by the other reg- 
ular polygons, up to the dodecagon or polygon of twelve sides. 

50. Proceeding with polygons, as before with rectangles, the 
only equal and regular polygons which can combine without 
leaving unfilled spaces between them, are the equilateral tri- 
angle, square and hexagon. Indeed, as the hexagon is itself 
composed of six equal equilateral triangles, we might say that 
the equilateral triangle and the square are the only indepen- 
dent figures that can so combine. Only, the equilateral tri- 
angle can combine in other ways than in hexagonal groups, so 
that, practically we may admit the three figures as separate. 

51. While combined rectangles constitute the more essential 
or useful members of many objects, combinations of various 
regular polygons, or other polygonal pieces founded upon them, 
may be made, as in PL XL 

Ex. 6. PI. XL, Fig. 1, a five pointed star. 

Ex. 7. PI. XL, Fig. 2, a clover-leaf pattern, where the centres of the circu- 
lar compartments are the vertices, a, 5, c, of the equilateral triangle abc. 
Ex. 8. PI. XL , Fig. 3, an' equally four armed cross. 
Ex. 9. PL XL, Fig. 4, an eight pointed star. 

52. An immense number of decorative designs, wholly or 
mostly geometrical, can be based upon tl\Q square, divided as in 



98 FKEE-HAND GEOMETRICAL DRAWING. 

PL XI., Fig. 3, into nine equal squares, or, as in Fig. 5, into 
sixteen • the latter division being founded on the number 2, the 
former on the number 3. 

52. In many current systems, the lines composing these designs 
would be located by considerations of distance, in some obvious 
systematic manner. But according to the principles of direc- 
tion, and of simple ratios between angles, as properly govern- 
ing geometrical design, these lines should have simple angular 
relations to each other. Thus, wishing, in PL XI., Fig. 3, to 
place a four-pointed star behind the cross, lines may be drawn 
from each corner of the large square, as at A, where the lines 
run to B and C, corners of the furthest arms of the cross. Or 
they may be drawn, as at D, to corners, E and F of the central 
small square, or again, so as to divide the distance C E or C II, 
in any given manner. These methods may seem sufficient, be- 
cause, after a fashion, they are definite and systematic, though 
the angles at A and D have no simple ratios to each other or to 
an angle of 90°. 

53. But it happens that definiteness and system can be had 
in another, and, according to the principles before established, 
a better way. Thus, the lines at d make angles of 30° with each 
other, giving, with the right angle at d, the ratio ^-. Again, the 
star lines at a include an angle of 45°, thus forming with 90° the 
ratio I-, while the external angles at a and c are of 22° : 30' and 
67° : 30' respectively, giving the ratios with 45° of \ and J, and 
with each other of \. The point at a is less clumsy than that 
at A ; that at d is more decidedly acute than that at D, if de- 
cision as to the acuteness be wanted ; or, if something nearly 
like the point D be desired, with harmonic angular ratios, it can 
be had by substituting 36° for the angle of 30° at d. This will 
give angles of 27° each side of it at d, and thus the ratios 

2.7. — 3. 2 7 — 3 • 36 — 2 
3 6 — 1 ? W — T 0" ? "9 0" — T' 

Here it is especially interesting to note, that if the star-point 
at A be considered well proportioned for a stout one, it is not 
so, on account of its principle of construction, but because, as 
shown by calculation, its angle is very nearly one of 54°, which 
bears the simple ratio, J- to 90°. Likewise if D be preferred to 
d, the secret of its superiority lies not in the manner of drawing 
its sides, but in the fact that its angle is very nearly one of 36°, 
which would bear to 90° the simple ratio, J- ; and would divide 



GEOMETRIC BEAUTY OF POLYGONS. GEOMETRICAL DESIGN. 99 

the right angle at D into the varied parts, 27°, 36°, 27°, instead 
of the three uniform ones, of 30° as at d. 

5-i. Again, PL XI., Fig. 5, may serve as a guide to many 
designs based on a sixteen-fold division of the primitive enclos- 
ing square, and made on the principle of simple ratios among 
the angles. The angles of the corner points are 36°, giving 
adjacent and alternate angles of 27°, and thence a ratio of J, 
and the ratio of 36 to 90 or f. The outer angles of the inter- 
mediate points are 60°, and their inner ones 90°, affording the 
ratio f. The obtuse lateral angles of the same points are 105° 
each, thus introducing, and with evident good effect, ratios of 
T 6 / y and T W or % and -f- in which the hitherto excluded num- 
ber, 7, appears. (See Note on Art. 20.) 

The angles of 120° and 153°, adjacent to each other, give the 
ratio 4-5-, so near f-jj- = £ that it may be called a tempered f, 
while the three angles of 105, 120, 135 give the ratios \ % -J, 
and f. 

Assembling now the ratios found, in progression, we have — 



* 



— 4 



T— o 

o — 



in which we miss only f from the continuous series, while we 
have in its place %, bearing to f the previous ratio § ; and -J, 
bearing to f the previous ratio -J. 

The modification of PL XL, Fig. 5, shown in Fig. 13, is pri- 




R* 



13 



marily afforded by enlarging the small corner squares, until 
they are exactly embraced, as at a and c, by the sides of the 



100 FREE-HAND GEOMETRICAL DRAWING. 

corner points. The arrangement thence suggests other lineSj 
which mav or may not be preferred to the less elaborate Fig. 
5, on PL XL 

Ex. 10-12. Draw PL XI., Fig. 3, uniformly; PL XL, Fig. 5; and com- 
plete Fig. 13 as here begun. 

55. Here again, having fully stated and explained the guid- 
ing principles, we will, in place of an extended series of copies, 
all essentially alike in being founded on fanciful relations of 
dista?ice, ask the pupil to exercise himself fully on the follow- 



Ex. 13. General Example. — Prepare several groups of squares, with three 
slightly separated squares in each group. 

Divide one square of each group into four equal squares, another into nine, 
and the other into sixteen ; in order to secure a regular or symmetrical figure 
in each case. Then, as in PL XI. , Fig. 3, place in like compartments of each 
square some combination of straight lines, located with reference to simple 
ratios between the angles which they make with each other, and between 
these angles and a right angle. 

56. PL XL, Fig. 6, shows how pentagons combine, leaving 
rhombus-formed openings between them. But the angles have 
simple harmonic relations, and give an agreeable figure. The 
angles of the pentagons being 108° each, the acute angles of 
the rhombuses are 36° each, giving the ratio ^, and the obtuse 
ones are 144°, with which 36° makes the ratio |-, and with 
which 108° makes the ratio f. 

57. Further geometrical decoration of each pentagon may be 
made in various ways, which will readily suggest themselves, 
under the guidance of our uniform principle of simple angular 
ratios. Thus, by drawing all of the diagonals joining alternate 
points of a pentagon, a five-pointed star will be formed, PL 
XL, Fig. 1, having a pentagon for its central body, on which 
stand the points. It is interesting to notice also, that the tri- 
angle, ABC, of 36°, 72°, 72°, and described in Art. 39 as re- 
presenting the third order of geometric beauty, founded upon 
the number 5, is not arbitrarily so described, since it is natur- 
ally derived from the regular pentagon, which by its five equal 
sides and angles, represents in geometry the numeral five in 
arithmetic. 

58. Regular hexagons combine without leaving vacancies be- 



GEOMETRIC BEAUTY OF POLYGONS. GEOMETRICAL DESIGN. 101 

fcween them, as will be readily seen on trial. They may be 
easily drawn by means of the equilateral triangles found by 
dividing the sides of a large equilateral triangle into the same 
number of equal parts, and drawing parallels to the sides, 
through the points of division. 

Ex. 14. Construct a group of equal regular hexagons. 

Octagons, whether regular or with alternate sides smaller 
than the intermediate ones, combine so as to leave square spaces 
between them, PL XL, Fig. 7. Here the angles of 90° and 
135° at any of the corners, give the simple harmonic ratio f. 

The size of the corner squares, Fig. 7, will be properly deter- 
mined by harmonic division of the sides of the original squares 
from which the octagons are formed. That is, the corners of 
the small squares should (25, 35) divide the sides of the original 
squares into parts having simple ratios to each other. 

We will here leave rectilinear combinations, having given 
the guiding principles and illustrations which may enable the 
learner to make any designs founded upon rectangles and re- 
gular polygons, so that they shall possess geometric beauty of 
form. 

59. The principal field for rectangular work will be found 
in the main divisions of build ino-s. That of various triangular 
and polygonal work will consist in subordinate features, bay- 
windows, summer houses, etc., and in geometrical decorations, 
such as wood or tile inlaid work, and other geometrical surface 
decoration. 

Ex. 15. Construct PI. XI., Fig. 6, and inscribe in each pentagon a five- 
pointed star, as in Fig. 1. Also make the central small pentagons, as abc, 
Fig. 1, black, and the star points lightly shaded. 

Ex. 16. Construct PI. XL, Fig. 7, with any additional geometrical decor- 
ation. 

Ex. 17. In Ex. 15, make the corner squares smaller, and a square on each 
side of each. A group of five equal small squares will thus be uniformly 
placed at each corner of the octagon, and oblong hexagons will be included 
by the combined squares and octagons. 

Ex. 18. Construct PI. XL, Fig. 8, as it is, then inverted, then turned 
right for left, and then inverted again ; making it two or three times as large 
as shown. This exercise, often repeated, both on paper and blackboard, will 
greatly aid in accurately estimating all the most important angles occurring 
in geometrical decoration, and in various positions. 



CHAPTER VI. 



CURVILINEAR GEOMETRIC BEAUTY. 



. Circles and Ellipses: 

60. The principle of unity, already denned (Arts. 6-8), to- 
gether with that of temperament (34), will here be remarkably 
exemplified by considering curves, not separately from the pre- 
ceding rectilinear figures, but as combined with them. 

61. The circle is a curve, all of whose points are at a uni- 
form distance, called the radius, from a fixed point within it, 
called its centre. 

The ellipse is a curve, such that the sum of the distances of 
each of its points from two fixed points within it is uniform, 
and equal to the longest line which can be drawn in the curve. 
These fixed points are called the foci of the curve. 

According to this definition, the curve, Fig. 14, may be de- 
scribed by a point P, moving so that the sum of its distances 







from two foci, F and F ; , is always the same, and equal to the 
longest chord, or transverse axis, AB, of the curve. Many 
differently shaped ellipses, wide or narrow, may thus be formed 
according to the distance apart of F and F', while AB remains 
of fixed length. Pins being fixed at F and F', and a firm 



CURVILINEAR GEOMETRIC BEAUTY. 



103 



thread of the length PFFT being placed around them, and a 
pencil point at P, this point, moved so as to keep the string 
stretched, will trace the ellipse. CD. perpendicular to AB at 
its middle point, O, is the shorter, minor, or conjugate axis. 

02. Harmonic relations of the triangle, square, and circle. — 
Fig. 15, represents a square, 2, 6, 10, 14, each of whose sides 

FVg. if. 



ft 


tsr 







i 




it, 


^^^i 


W^ 


/ 










*\W^\ / 


li 


w/^ 






b\ V\ 




^A* 


*\ / 








#2- 


X / 






c \ 

d \ , 




V 


It 


7 






e / \ 


X 


A 




^_\ 


/^^ 


/< 


3 


J 




r j 


? 





s 



is divided into four equal parts. Lines parallel to adjacent 
sides of the square, through these points of division, divide the 
area of the square into sixteen equal square parts, as the peri- 
meter is already divided into sixteen linear parts. 

If, now, a circle be inscribed in this square, the lines just 
described will divide its circumference into twelve equal parts, 
as numbered in the figure, and radii from the points of divi- 



104: FREE-HAND GEOMETRICAL DRAWING. 

sion will divide the area of the circle into twelve equal parts. 
Here we have the ratio § between both the linear and area 
divisions. 

The inscribed figures are also remarkable. 

Remembering that an angle at the circumference of a circle 
is measured by half the arc of that circumference between its 
sides : 

1°. The inscribed triangle, 3, 9, 0, and therefore, also, its 
half, 9, c, 0, is one of 45°, 45°, 90°, that is, one of the first 
order (37). 

2°. The triangle 8, 4, is an equilateral triangle, whose half, 
8, d, 0, is therefore one of 30°, 60°, 90°, that is, one of the 
second order (38). 

3°. The triangle 10, 2, is one of 30°, 30°, 120°, whose half, 

10, h, 0, is therefore again one of 30°, 60°, 90°. 

The foregoing are the most remarkable, in connection with 
the subsequent figures, though it is interesting to note the fol- 
lowing, also. 

4°. The triangle 11, 1, is one of 15°, 15°, 150°, whose half, 

11, a, 0, is one of 15°, 75°, 90°, giving the ratios -J, -J-, | (40). 
5°. The triangle 7, 5, is one of 30°, 75°, 75°, whose half, 7^0, 

therefore gives again, but as in (2°) with the longest base verti- 
cal, a triangle of 15°, 75°, 90°, w T hose ratios are -J-, -J-, f, as before. 
Finally, drawing the chords 1, 11 and 5, 7, we have the rect- 
angle 1, 5, 7, 11, belonging to the second order of symmetry 
(38) in that its half, the triangle 1, 5, 7, is one of 30°, 60°, 90°. 

63. Harmonic relations of ellipses. The ellipse, being so to 
speak a somewhat monotonous curve, owing to its double sym- 
metry (Part I., Ch. VII.) which makes its four quarters alike, 
it is less valuable for many decorative purposes than the egg- 
formed curves, which will be described further on. 

We shall therefore here treat this curve and its relations, 
less fully than Hay has done, but more completely, so far as 
the treatment goes ; and with much more elementary demon- 
strations of the properties noted. 

In Fig. 16, the rectangle 2', 6', 10', 14', is of the same pro- 
portions as the inscribed one, 1, 5, 7, 11, in Fig. 15, but larger, 
its longer side being equal to aside of the circumscribing square 
in Fig. 15, while its half 2', 6', 10', is a triangle of 30°, 60°, 90°. 



CI K\ ILINEAR GEOMETRIC BEAUTY. 



105 



By the definition of the ellipse (Art. CI), describe arcs with 
4' and 12'um the rectangle) as centres, and the half side, 4', 2', 
as a radius, and they will meet on the line O'c'8'at the foci, 
F and F'. Having the foci, and axes, (V, 6', and 3', 9', the curve 
may be drawn by Art. (61), or otherwise, as most convenient. 




So much being done, divide the sides of the rectangle, each 
into four equal parts, and join the points of division as in the 
square, which, as before, will divide the rectangle into sixteen 
equal parts, and the circumference of the ellipse into twelve 
parts, which, however, are not equal. 

64. We shall now find, by applying a protractor* the fol- 
lowing remarkable results ; naming the inscribed triangles in 
the same order as for the square. 

1°. The right angled triangle 9, o of 45°, 45°, 90°, is 

* A graduated semi-circle for measuring angles in degrees. 
5* 



106 FREE-HAND GEOMETRICAL DRAWING. 

•transformed into the semi-equilateral triangle, 9' c' 0', of 30°, 
60°, 90°, that is, from one of the first, to one of the second order 
of symmetry. 

2°. The triangle 8 d 0, of 30°, 60°, 90°, is transformed into 
the triangle 8'd'O'which is sensibly one of 18°, 72°, 90°, that 
is, from one of the second to one of the third order of sym- 
metry. 

3. The triangle 10 b is transformed into 10' V 0' of 45°, 45°, 
90°, that is from the second, to the^r,^ order of symmetry. 

4°. Taking the supplementary triangles ; 11 a is replaced 
by 11' a' 0' which is almost exactly one of 18°, 72°, 90°. 

5°. Also, 7^0 is transformed into T'e'O', a triangle sensibly 
of 9°, 81°, 90°, giving the ratios |, T V, T %- 

65. Demonstrations* We shall uniformly call the radius 
of the circle, Fig. 15 (equal to c' 0', Fig. 16) = 1. Then— 

1°. The triangle 9' c' 0'. The rectangle, Fig. 16, is, by con- 
struction, one whose triangular half, 2', 6', 10', is a triangle of 
30°, 60°, 90°. The triangular quarter, 9' c' 0', of the rectangle, 
is evidently of like proportions, and hence is exactly a triangle 
of 30°, 60°, 90°. 

To find the value of c' 9', refer to the triangle ce 7, Fig. 15, 
also evidently similar to the preceding, and we have 

ce :e7 :: c'0' : c'W 
that is, Ws :i::l : c'9'. 
\t 1 



Hence c'9': 



Ws V3 



2°. The triangle 10' b' 0'. Here it is necessary to explain, 
first, that a perpendicular as b 10, or b' 10', from a point of the 
circumference to a diameter or axis, of a circle, or ellipse, is 
called an ordinate to that diameter. Also, if the circle, Fig. 15, 
be revolved about the diameter 0, 6, until the diameter 3 c 9 be 
inclined to the paper so as to appear of the length 3' c' 9, the 
ordinate b' 10' will be parallel in space to c' 9', both being 
equally inclined to the paper at V and c' . Perpendiculars, 



* These, given by Hay in an appendix, and by the methods of analytical geo- 
metry, suited therefore only to advanced students, are here given arithmeti- 
cally in a way suited to pupils acquainted with the elements of geometry- 



CURVILINEAR GEOMETRIC BEAUTY. 107 

from 10' and 9' in space, to the paper, will be parallel, and we 
shall thus have the two similar triangles standing on b' 10' and 
c 9', which when placed together, so that b' 10' falls on c' 9', 
and then turned down into the paper will appear as in Fig. 17. 



F,> it 




id n 



Then we evidently have b' 10' : ¥ 10': : d 9' : c' 9. That is, 
see Figs. 15, 16, the ordinate b' 10' of an ellipse, is to the cor- 
responding ordinate, b 10, of the circle described on the longer 
axis of the ellipse as a diameter, as the semi-conjugate axis, 
c'9', of the ellipse, is to its semi-transverse axis, cd (=c / / > 
Fig. 16). 

But b 10 = -J\/3 and c' 9' = — y=, hence, only changing the 

order of the proportion, 

cd: c'9'::bl0:b'10'i 



that is 



l:^g::iV3:6'10. 



Therefore b' 10 = % -Ll = ± = b' 0'. 

Vs 

Hence the triangle 10' b' 0' is exactly one of 45°, 45°, 90°. 

3°. The triangle 8' d' 0'. Here d' 0' = f and 8' d' = \. 
Then to find the distance c'jp' corresponding to 8' d', when d' 0' 
is reduced to c' 0' or 1, we have 

f :l::i:*V, 

whence c' p f = f- = J. 

I 



108 



FREE-HAND GEOMETRICAL DRAWING. 



Now from a table of natural tangents* (see works on trigo- 
nometry or surveying), we find that i is the natural tangent of 
18° : 26' : , or more exactly, of 18° : 26' : 6". 

The triangle 8' d' 0' is thus almost exactly one of 18°, 72°, 
90°, the slight difference being an illustration of temperament, 
(Art. 34), in this case the modification of the perfect triangle 
by the constraining effect of combining it with an ellipse, un- 
der a rigid system, that of the sixteen-fold equal division of the 
circumscribing rectangle. 

4°. The triangle T e' 0'. Here c e' = oe, Fig. 15, (65, 2.°) 



But ce = V(c7f—(e7f = Vl-i = 
Then V = 1 +W3 and e' 7' = \d 9' 



1 



To find c' q\ or e' 7' reduced to correspond with 0' c', or 1, 
we have, 

0'e':0'c'::e'7' :c'q' 

or 1 + WS :l::o7^:cy 



whence c r q' 



2 1/3 



1 

w 



ii/3 



2+|/3 2^/3+3 



0.1547 



which is the natural tangent of 8°: 48'— , showing that 7' e f f 
is very nearly a triangle of 9°, 81°, 90°. 



* If the hypothenuse, O B, Fig. 18, of a right-angled triangle be taken as a 
radius, as shown by the arc B5, and called 1, the other sides will be called, 
AB, the natural sine, and OA the natural cosine of the angle O, at the centre. 



Fig. 13 




But if the side, OA, of the right angle, be taken as such radius, as indicated 
by the arc Aa, the other side, AB, will be the natural tangent of the same 
angle. 



ft 



PLXI. 




C A 



:^ 


c 


-7 




F 


E 
H 


> 




A 



F.g.3. 




F.g. 4. 





<« 








^- 1 ' •A' 5 " rf"' 






\;x 3 " 








./ffiy 































F/ g ..r 



CURVILINEAR GEOMETRIC BEAUTY. 109 

5°. The triangle 11' a' 0'. Here*. 0' a' = 1 — \ y/W and 

a' 11' = — — . Then to find the angle at 0' to a radius of 1, 

i 
we must proceed as in 4°, whence 1— -£ 1/3 : 1 : : k~?t : tang. 

a'O'ir. 

1 1 



That is, tang, a' 0' IV - " 2 ^ s 




— 2.15425 = the natural tangent of 65° : 6'+. 

angle 0' a' W is not a very close approximation to one of 

22° : 30', 67° : 30', 90°. 

66. Proceeding in like manner, beginning with a rectangle 
composed of two triangles of the third order of symmetry 18°, 
72°, 90° (Art. 39), we should find like curious and interesting 
results. Tims 0', 2', 10', Fig. 16, will then become an equila- 
teral triangle, or its half, one of 30°, 60°, 90° ; 0', 3', d will 
become a triangle of 18°, 72°, 90°, and 0' 4' d\ one of 10°, 80°, 
90°, etc. And, in general, the triangle on 2', 10' of a series of 
ellipses thus formed, will be of the same form as the one on 
3' o' 9' of the next preceding ellipse, and so on ; the one on 4' 8' 
being of the next higher degree in the series of triangles, and 
the one whose half determines the form of the circumscribing 
rectangle of the next ellipse. 

Elliptical Designs. 

These may be formed by combining ellipses, as indicated in 
the following examples, and in other ways which the pupil 
may invent. (See Part I., p. 40.) 

Ex. 19. Construct the next three ellipses to Fig. 16, indicated in the last 
article. 

Ex. 20. Combine the circle and the different ellipses, by pairs, replacing 
the left band half of Fig. 15, by the left hand half of Fig. 16, for example. 

Ex. 21. Proceed in like manner with either Figs. 15 or 16, and either of 
the ellipses of Ex. 19. 

Ex. 22. Making Fig. 16 larger, so that 3', 9' shall be equal to 3, 9 in Fig. 15, 
replace the lower half of Fig. 15 by that of the new Fig. 16. Do the like with 
the ellipses of Ex. 19. 



CHAPTER VII. 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. 



Natural and Artificial Curves. 

67. A natural curve is formed so that every point of it is 
located by one and the same law of construction ; as when a 
circle is drawn with a pair of dividers carrying a penc.l, whose 
describing point moves in obedience to the one law of always 
being at a uniform distance from a fixed point. 

68. A curve is also natural, when it arises by cutting the 
surface of some simple body — also formed according to some 
one law — by a plane or by some other simple surface. Thus, a 
cone is formed by revolving a right-angled triangle about 
either of the sides of the right angle ; and the curves which 
arise by cutting its curved surface by any plane are natural 
curves. 

69. An artificial curve, on the contrary, is built up of sepa- 
rate arcs, of circles of different radii, or of some other kind of 
curve, placed tangent to each other. 

70. Thus the natural ellipse, Fig. 14, is described, according 
to Art. (61) by a continuous movement of one kind. 




71. The artificial ellipse may be composed in various ways 
of four or more circular arcs, one way being shown in Fig. 19, 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. Ill 

where C, O, P, Q, are the centres of the two pairs of arcs, as 
ab, and be, there used. But the resulting figures are all more 
or less unsightly curves, in that they appear swelled at the 
middle of each arc, and flattened at the junctions of the differ- 
ent arcs, owing to the sudden and great changes of radius 
which occur at the latter points. 

The Egg-Form, or Oval. 

72. This curve invites examination hy its constant occur- 
ence in Nature, and its great beauty. But in turning from 
Nature to Geometry, we find no mention made of egg-formed 
curves in ordinary treatises, while in the more extended works 
on the higher curves, we find them occasionally as detached 
portions of curves having a complex law of construction, and 
which consist of two or more separate brandies. 

73. This absence of egg-curves, constructed by any simple 
rule, may account for the customary substitution, in practical 
drawing books, of artificial egg-curves, composed of circular 




arcs, as in Fig. 20, where A D B is a semi-circle ; A E and B Gl- 
are arcs described from B and A, respectively as centres, C the 
intersection of their radii, being the centre of the arc E F G. 

74. Hay's composite ellipse. — Still, it is desirable, both for 
the sake of conformity to Nature, and of completing a system 
of geometric beauty, that an oval curve should be found, as 
naturally and intimately associated with the various harmonic 



112 FREE-HAND GEOMETRICAL DRAWING. 

triangles (37-39) as the circle is with the square, or the ellipse 
with the rectangle. 

Hay perceived and stated this want, and, in his search to 
supply it, devised the egg-curve which he called the composite 
ellipse, Fig. 21. Here, the points A, B, C, are foci, and D is 
the describing point. That is, pins being set at A, B, C, D 
and an inelastic string tied tightly around the four, the pin at 
D is then replaced by a moving pencil point, as at P, Fig. 14, 
and this pencil, moved so as to keep the string tense, will de- 
scribe the egg-curve shown in Fig. 21. Comparing this with 
Fig. 20, the curve will evidently consist of several elliptical 




arcs ; first rtiDn, with B and C for its foci ; second no, with 
A and B for foci ; third, op, with A and C for foci ; next, the 
short arc pq with B and C again as foci, etc. The curves thus 
formed are divided by Hay into five " classes," according as 
the angle at D is 90°, 108°, 120°, 135°, 144°, these all having 
simple ratios to 180°. Each class is then divided into as many 
" degrees " as there may be angles chosen at A having simple 
ratios to the fixed class angle at D. 

Thus a great variety of egg-forms, or ovals, including many ^ 
oblate ones, are formed ; as when, for example, the angles at 
A and D are made 90° and 135° respectively. All these can 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. 113 

readily be traced, by the student who would gratify his curiosity 
to know how they would appear. 

75. Objections. All this seems simple and ingenious, but — 
First, the guiding idea of simple ratios between the angles of 
the same right-angled triangle as A E B is partly, at least, 
abandoned. Thus Fig. 21, 13^=^^=^ a very odd ratio. 

Toi 

Second. The curve is artificial, and not natural, as much so 
as are Figs. 19 and 20, in that it is built up of arcs of another 
kind of curve. 

Third, accordingly, pottery, and other designs formed with 
it, fail to perfectly satisfy the eve of persons of taste, not 
knowing how these designs were formed. That is, these de- 
signs imperfectly realized a good general idea, just to the extent 
that these curves were artificial, and also perhaps inappropri- 
ately combined. 

76. The natural method. Happily, there remains another 
path which promises to lead to the desired curve. It is to be 
remembered that curves never occur in nature as separate in- 
dependent things, but only as the outlines or sections of the 
surfaces of bodies. Hence, in place of seeking among collec- 
tions of all the curves which accident or the wit of man have 
suggested, we shall seek some surface whose sections will be 
egg-curves or ovals. 

One of the best for our purpose is the following. 

77. Let A C B, Fig. 22, represent a semi-circle, standing per- 
pendicularly to the paper, and thus appearing to coincide w T ith 
its diameter A B, which is supposed to be in the surface of the 
paper. Then A O B will represent this semi-circle, after re- 
volving it over into the paper, about its fixed diameter A B as 
an axis. Next, let A O B be an isosceles right-angled triangle, 
inscribed in this semi-circle, and produce its sides at pleasure. 
Transform this triangle into any other, as a O b, in which one 
vertex shall remain at O, and the altitude shall still coincide 
with O C, and make the angles at a and b with any simple ratio 
to each other. In the figure, this ratio is -J-, the angles at a and 
b being respectively of 30° and 60°. 

So much being done, mark the points b, C, a on the edge of 
a slip of paper, and then adjust the position of the slip by trial. 



114 FREE-HAND GEOMETRICAL DRAWING. 

which can easily be done, so that the distances a C and b C shall 
be included, as shown at a x <?, and .b 1 c„ between the lines O A 
and O C, and O B and O C, the lines distinguished by stars. 




Then, assuming any points, as n and p, equidistant from C for 
convenience, draw nm&n&pr perpendicular to A B ; and draw 
O n and Ojy, and note their intersections, n x and^ with a x b v 
Also note g x . Finally, at n x e l9 p, draw perpendiculars to a x b 19 
and make n 1 ?n 1 = n?ri; c 1 1 = C O ; p x r 1 -=jpr. Then the curve 
a i Oi b L will be that semi-oval which circumscribes the triangle 
a, O x b r of 30°, 60°, 90° derived from the triangle A O B as de- 
scribed, that is, so that O D bisects the angle A O B. 

78. Variations from Fig. 22. These may be made in three 
ways.* 

1°. By dividing the angle A O B equally, or otherwise. 

2°. By making the ratio of the angles, a and b, of the trans- 
formed triangle a Ob, either the same as, or different from that 
of the two angles into which A O B is divided. 

3°. By making the transformed triangle aOb(oraS b — see 
Art. 79) right angled, or not, at O. 

79. Illustrations. Of Figs. 23 to 30, inclusive, each ill us- 



* I find by trial, that, other things being- the same, the position of 0, aa 
the centre of the radials 0«i, On, Op, etc., does not affect the form of the 
oval. 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. 



115 




116 



FREE-HAND GEOMETRICAL DRAWING. 



trates one or more of these variations. Similar points having 
the same letters on all of them, and the construction of all 
being the same, with one exception, next mentioned, repeated 
description is unnecessary. 

But note, that when the angle A O B is divided unequally, 
as in Figs. 23, 24, etc., the vertex of the transformed triangle 
moves from O to S, the summit of a perpendicular to A B from 
its intersection. 6, with the dividing line O D, which then no 
longer coincides with O C. 




445: 

en 






,.-'■; «mj 



.Returning to the illustrations ; In Fig. 23, a S h is a right 
angle, while in Fig. 24, it is not. In Fig. 22, A O B is divided 
equally, and in Figs. 23, 24, etc., it is not ; the degrees in each 
division being marked at D. In Fig. 23, the angles at a and b 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. 



117 



have the same ratio as the parts of AOB, while in Fig. 24, they 
do not. 

SO. Pairs of orals. Notice, farther, that whenever AOB 
is divided unequally by OD, we can obtain two ovals from 
the same transformed triangle a S b ; according as either seg- 
ment as a e, or b e, of its base is inscribed in the larger, or small- 
er of die angles into which AOBis divided, as at b t e± = be, 
or <22 ^5 = be, in Figs. (23 — 2G). 












\ 

jus: 



ru«x 



I 



81. Results of the variations. — 1. The more acute the angles 
at a and &, the longer, and more nearly pointed, the oval be- 
comes. 

2°. The greater the difference between the angles at a and b, 
the greater is the difference in the form of the two ends of the 
oval. 

3°. But when the shorter segment be of the base ah is in- 
scribed in the larger of the two unequal angles into which 
AOB may be divided, the more unequal these angles, the 
more acute is the oval at the end at the acute angle, and obtuse 
at the opposite end, as is successively more and more strikingly 
shown by the oval a 2 o 2 b 2 in Figs. 23, 24, 25. 

82. To make a flat oval — one broader than it is long — the 



118 



FEEE-HAND GEOMETEICAZ, DRAWING. 



t ^3S<i 







Flg.*7. j* 




s 







v _AI\ N 
s I \ 

\ I \ x 

Fig-. n- s \ 



K 



CURVILINEAR GEOMETRIC BEAUTY. OVALS. 



119 



base ah, Figs. 29, 30, of the transformed triangle, must be 
shorter than the greatest width, 2 O C, or 2 S e, of the oval. 
The axis, a x b u of the oval, being thus brought nearer to O, the 




Fig. 29 



y--rSi 







X ' 

Fig. 90. 

semi-circle A O B should be larger, to give an oval ol the 
desired size. 

S3. Only half ovals are shown, in order to avoid confusing 



120 FREE-HAND GEOMETRICAL DRAWING. 

the figures. They will exhibit themselves better by completing 
them ; the two halves on each side of a x h u or a 2 b 2 , being just 
alike in all the figures. 

With these explanations and illustrations, pupils and practi- 
cal designers will be put in possession of the means of con- 
structing ovals of every possible variety of form, and based 
upon definite angular relations. 

Any oval can then, if desired, be enlarged or diminished in 
size, without altering its form, and without the labor of entirely 
reconstructing it, by either of the methods explained in Part 
L, PL IL, Figs. 7, 8, 9.* 



Industrial Applications. 

84. The industrial applications of the ovals now described, 
both to objects of utility and beauty, are so numerous that, for 
want of space, we can only name some of them, in connection 
with the principles governing their design, and illustrated by a 
few sketches. But more than this is unnecessary, for as the de- 
signer is furnished with the necessary principles to guide him, 
he is made independent of multiplied copies to be merely imi- 
tated without further thought. 

Among the most important of such objects, are those of 
pottery and glass; as pitchers, bowls, vases, goblets, fruit-dishes, 
gas-shades, etc. ; and of metal, as tea-sets^ butter dishes, vases, 
lamps, urns, etc. ; also architectural ornaments, mouldings, 
railings etc. ; and church, school, hall and house furniture. 

85. Free application. — The ear is the best instrument for 
testing the musical beauty which results from the combination 
of musical chords, separately agreeable, into a piece of music. 
Likewise the eye is the best instrument for testing the beauty 



* The advanced student, if acquainted with Descriptive Geometry, will 
recognize the surface from which these ovals are cut as a right conoid, whose 
directrices are the semi-circle ACB, and a vertical at O, equal to AC, and 
whose elements are all parallel to the paper. 

Besides the conoid, egg forms can be cut from the " annular torus," a ring 
of circular or elliptical section, by planes parallel to its axis, and cutting it in 
two curves. 






:i 



( I KVILINEAR GEOMETRIC BEAUTY. OVALS. 121 

of compound forms, made by combining separately pleasing 
elementary forms. 

Moreover, as we have seen, very many ovals can inscribe the 
same harmonic triangle. Thus, in various ways, it is evident, 
that, in the higher forms of curvilinear beauty, we depart 
further and further from the domain of rules alone, and that 
the prineiple of freedom (Art. 11) or intuitive perception of 
beauty prevails; just as in music, a piece may be composed, 
strictly according to rules, and yet be utterly destitute of 
beauty, while another maybe very beautiful, and yet the secret 
of its beauty be inexpressible by any rule. 

Hence, for all purposes of application of the ovals here de- 
scribed, it is sufficient to direct the designer to construct a large 
number of them on card-board, classifying them according to 
the distinctions given in Art. (78), and then to combine them 
tangentially to each other, or intersecting each other also, as in 
PL XII., until the outline formed by their combination sat- 
isfies his eye ; or is pronounced beautiful by persons of taste, 
independently of each other, and, better still, unbiassed by 
a knowledge of the method by which the design was pro- 
duced. 

And here note, that angles formed by the meeting of curves, 
are, for purposes of comparison, to be estimated by the tan- 
gents to those curves at the point of intersection of the curves. 

86. Application, governed by rules. The ovals here described 
are so graceful, that they combine together in graceful forms 
almost as readily as different plant leaves do in a bouquet. 
Still, if ornamental forms, like those of PL XII., are to be de- 
signed in a strictly systematic manner, instead of by merely 
satisfying the eye by trial, three points may be kept in mind 
while making the design. 

First ; If different ovals are to be used in the same design, 
those may be chosen in which the angles of each of the inscribed 
triangles form harmonious, or simple ratios with those of the 
others. 

Second ; The angles made by the axes of auxiliary ovals, with 
the vertical axis of the entire figure, as in the neck and foot of 
Pig. 1, may have a simple ratio to 90°. 

Third ; According to Art. (35) the axis of the figure may be 
6 



122 FREE-HAND GEOMETRICAL DRAWING. 

divided by the different members of the entire design, into 
segments having simple ratios to the whole height. 

87. The designs on PI. XII., were mostly formed by the first 
method (85) that of satisfying the eye by trial. Yet the seg- 
ments of the heights are, generally, very nearly if not exactly 
simple ratios of the whole heights. These designs are not 
offered as models of oval composition, but only to indicate the 
manner of combining ovals in forming regular objects with 
curved outlines. 

Fig. 1, for example, is composed of the oval a x s x &, of Fig. 25, 
for its body, with a part of a 2 s 2 b 2 , Fig. 23, for its neck, and of 
the end at a x of a x s x b„ Fig. 22, for its foot. From this begin- 
ning, innumerable minor modifications can be made by the 
pupil or designer, by taking variously proportioned ovals for 
the body, and various portions of different ovals for the neck 
and foot, until the most satisfactory forms shall be attained. 
In like manner, each of the following examples may be vari- 
ously modified. 

88. PL XII., Fig. 2, represents a glass fruit dish, the body 
composed of the lower portion of Fig. 30, ending a little above 
its line c, <?„ of greatest width ; and the foot composed of an 
arc of a a s 9 'b 3 , Fig. 24, in the vicinity of s r Slight variations in 
the fashion of the body would make its brim just on the line, 
c i °\i °f greatest width, or a little below that line. 

The top of the dish, being of less diameter than the body 
below, expresses reserve. When, as in the last modification 
mentioned, the top diameter is greatest, generous freedom is 
expressed. Or, leaving sentiment for utility, the first form is 
better adapted to carry fluid contents without spilling, and the 
second, to supporting a pyramid of fruit. 

89. PL XII., Fig. 3, represents a garden vase. The outline 
of the upper member is an arc of the oval in Fig. 22, from O t 
towards a- 19 and is superior in two ways to an outline composed, 
as is sometimes done, of a straight line tangent to a circular 
arc. First, it is wholly curved. Second, the curvature con- 
stantly varies, instead of being uniform, that is, monotonous, as 
in a circular arc. 

The middle member is composed of the larger segment of a 
smaller oval of the same form as that of Fig. 29, and is much 






CURVILINEAR GEOMETRIC BEAUTY. OVALS. 123 

finer than a simple semi-circle, though the difference is not 
great on the scale of the figure. 

The lower member is composed of various mouldings, all the 
curved portions of which would, when of full size, be composed 
of arcs at the tip of some of the more pointed ovals. 

90. PI. XIL, Fig. 4, is a two-handled jug, in which, to secure 
flowing combined outlines, and an absence of straight lines, the 
form of the actual piece of pottery from which the figure was 
taken, was modified by giving a very slight curvature to the 
handles, taking for this purpose the straightest portion of the 
oval a^ ? 2 b 2 of Fig. 24. 

Also the decision expressed by an exact right angle is secured 
by taking the top outline of the oval a 2 s 2 b 2 of Fig. 26, for the 
curve of the top of the figure. A portion of a new oval, not 
shown in any of the diagrams, forms the body of the jug. It 
was formed from a circle of a diameter equal to the greatest 
diameter of the jug, with the angle A OB bisected, and the in- 
scribed triangle of the half oval having base angles, a and b, of 
10° and of 60°. 

91. PI. XII., Fig. 5, represents a portable gas-light, in which 
the outline of the shade consists of a little more than the most 
flattened half, o 1 s x J 1? of the small oval similar to Fig. 29, the 
exact half, ending at the double lines near the top. The stand- 
ard is formed of an arc of the acute oval used in Fig. 4, but 
broken, to secure shadow and variety, by the two rings. 
Strictly, and when of full size, the moulding of these rings 
should be in ovals. 

92. PI. XIL, Fig. 6, represents a fruit dish wholly composed 
of arcs of the oval, of Fig. 30. The flatter half, enlarged a tri- 
fle, is taken for the body of the dish, terminating on the line 
c 1 s 1 of greatest width of the oval. The standard is composed 
of nearly the whole of the outline of the more convex half, a^o^ 
of the same oval. 



The Method by Co-ordinates. 

93. Compound, or waving curves may be sketched by a 
method suggested by that of finding the location of a stream, or 
other irregular line, in a survey, viz., by distances to the given 



124 



FREE-HAND GEOMETRICAL DRAWEE. 



line, measured perpendicularly from points at given distance 
apart on a fixed straight line of reference, as ah in Fig. 31. 













cp^/ 




' 




***& 


a 


6 






F% 


* 31. 





But in applying this method to the free design of some ideal 
line of beauty, we are no longer bound by given distances either 
on, or from ab, but can, according to the principles of this Part 
III., substitute angular ratios for them. 

Thus, in Fig. 32, the left hand profile, A, is determined by 
co-ordinate distances. The height 04 is divided by trial into 





fig. a*.- 



four equal parts, then 11 equals one of those parts, the next 
equals half of 01 ; the next, one-fourth of 01 ; and the top 
width is three-fourths of 01. This seems simple and systematic, 
but the result is less pleasing than B, or than Fig. 33, which it 
somewhat resembles. And we venture to say that any design, 
made like A, Fig. 32, by related distances, will only happen to 
be pleasing, as a result of repeated trials, while forms like B, 
and Fig. 33, composed of arcs of ovals, (85, 86) will almost 
always be graceful. 



CURVILINEAR GECttfETRIC BEAUTY. OVALS. 



125 



Fig. 33 is composed of an arc of the side of a 1 0, b x , Fig. 22, 
from A to B, and of the pointed end of a 1 s, b iy Fig. 26. It be- 




longs to that class of vase forms, which have a tapering neck 
and a somewhat sharply curved body, and is decidedly superior 
in configuration to like forms composed of arcs of composite 
ellipses. 



Ex. 



23. 

saucer. 

Ex. 24. 

Ex. 25. 

Ex. 26. 



Apply the principles of this chapter to the designing of a cup and 



Design, likewise, a garden urn or vase. 

Do. A fountain. 

Do. A summer-house, applying the principles of Chapters IV. 
and V. , to the rectilinear parts. 
Ex. 27. Do. A library table. 
Ex. 28. Do. A parlor stove. 
Ex. 29. Do. A flower garden. 
Ex. 30. Do. A pulpit. 



CHAPTER VIII. 

GEOMETRIC SYMBOLISM. 

Definitions, and General Illustrations. 

9<±. Among the elements of geometric beauty, but of a very 
different kind from those of harmonious proportion thus far 
explained, is the symbolism of geometrical figures, or the anal- 
ogies between some of their properties and certain elements of 
life. 

Examples of such analogies may here form an appropriate 
conclusion. They are generally expressed by the words, type, 
emblem, symbol, of which the last only will be particularly con- 
sidered. 

95. A symbol is anything apparent to sense, which yet, of 
itself, naturally expresses, represents, or suggests to the mind 
some truth of life ; the natural counterpart in the world of 
matter, of something corresponding to it in the world of mind. 

In this natural correspondence, a symbol is quite different 
from an emblem, or a type, as may be sufficiently seen by re- 
flection on the common use of the words. Thus every one says, 
" the national emblem" speaking of his country's flag, but not 
the national symbol. Here, the connection between the thing 
and the thought is dependent on association, and mutual agree- 
ment, and not on inherent natural correspondence and may be 
equally strong, whatever the thing chosen may be. 

A type belongs to the same general form of existence as the 
thing typified. It is a part, taken as a representative of the 
whole ; a specimen, as the representative of a class ; a lower 
form, as a representative of a higher form of being or action of 
the same kind. 

96. To illustrate : The mingled verdure and bloom of spring, 
are symbols of the freshness, modesty and promise of un per- 
verted youth. The tints and fruits of autumn, or a sunset in 



GEOMETRIC SYMBOLISM. 127 

crimson and golden light, are symbols of the close of a worthy, 
or a splendid career. 

A monument is an emblem of departed greatness. A "broken 
monument is a symbol of a broken life. The American flag is 
an emblem of the nation's life. Its rivers are the symbol of the 
scale of its life, its ideas, and its actions. Its best, and its 
worst, treatment of the Indians, are types of its highest and of 
its lowest humanity seen in all other relations of life. 

Again : Water, by its properties, is a tyjje of fluids generally. 
The ocean is of itself, because of its apparent boundlessness, a 
symbol of eternity. 

The oak, with its mighty and horizontal arms, is a symbol of 
self-sufficient rough and rugged strength, and independence. 
The elm is a symbol of united strength and grace, and thus of 
culture. Hence, apart from practical convenience, the avenues 
of cultured towns are appropriately lined with elms, rather than 
with oaks. 

With the idea of symbols thus awakened, the following ex- 
amples of geometric symbols will suffice to lead the mind into 
action on the subject. 

Geometric Illustrations. 

97. A straight line is the symbol of repose, monotony, per- 
manence and deadness. It is so by reason of its monotony of 
form, in having but one unchanging direction. It is therefore 
adapted to situations where repose, in the shape of fixedness or 
permanence, is natural or desirable. 

Thus, in the fervent tropical heats of a land like Egypt, where 
vigorous activity is to be dreaded, and the repose of utter inac- 
tion courted, the main outlines of the buildings, naturally and 
forcibly express these facts by the free use of straight lines, 
and these, as the boundaries of most massive and heavily pro- 
portioned forms. Stout and short vertical columns, mile-long 
avenues of bolt upright figures, with folded arms and all facing 
alike, and the immense horizontal bases of the pyramids, and 
the lines of the immense stones which compose them, all illus- 
trate this. 

Also, in foundations generally, where permanence is most 
desirable, the main lines are mostly straight and horizontal. 



128 FREE-HAND GEOMETRICAL DRAWING. 

But in a church, the multitudinous flowing and uptending 
lines should only express the endlessly varied, yet ouly beauti- 
ful and elevated, individual and associated life, that should, 
visibly, centre in, and flow from the stirring exercises and ac- 
tivities within it. 

98. The circle is a symbol of monotonous routine, and hence, 
as a symbol of eternity, represents only a dormant, unprogres- 
sive one. It is thus, by reason of its single centre and uniform 
distance from that to the circumference, and its consequent uni- 
form rate of variation of direction at all points, and its per- 
petual return to the same point of beginning. 

Hence it is peculiarly appropriate that a nation fallen into a 
state of decay or lethargy, and whose earthly life might then 
be largely expressed by the stiff, dead straightness of a right 
line, should adopt the circle as its symbol of eternity, an eter- 
nity of endless dull repetitions of one unvarying round. "One 
unvarying round," is just what the circle sensibly is, and it is 
therefore the natural symbol of a life made up of routine in 
one unvarying round. 





Fig. 34. Fig. 35. 

Again, life is either sensual or spiritual ; and, in a given 
amount of it, as the one prevails, the other is wanting. Now 
monotony of life indicates absence of thought-activity, and 
hence, secondarily, the circle as the symbol of monotonous rou- 
tine, unenlivened by varied thought, is also a symbol of sensu- 
ous, more than of intellectual existence. Hence the Romans, 
who were a grosser, and more materialistic people than the 
Greeks, spontaneously as it were, made great use of the circle 
in their architecture, while the Greeks rejected it. 

Thus the coarseness of the compound circular moulding, Fig. 



GEOMETRIC SYMBOLISM. 



120 



3-i, is apparent in contrast with that of the freely varied prin- 
cipal curve of Fig. 35, whose quick terminal curves, with the 
more uniform portion, included between them, readily express 
early entrance upon a prolonged career of excellence, promptly 
closed where its work is done. 



99. The ellipse being only the general form, of which the cir- 
cle is a particular case, it is not expressive of anything radically 
different from what is symbolized by the circle. Its continually 
varying rate of curvature expresses more of varied life than the 
circle does. Also its two foci, representing a two-fold govern- 
ing purpose, or idea, or all-engaging pursuit, give more of life 
to it as a symbol. 

As contrasted with a circle, for a window, its compression in 
one direction may make it expressive of partly constrained or 
contracted, rather than of full-orbed and equally all-embracing 
life and character. Hence elliptical topped windows, for ex- 
ample, are less frequent and pleasing than semi-circular topped 
ones. 

100. Quite otherwise from the foregoing is it with the hyper- 
bola, which is sufficiently defined for present purposes by say- 




Fig. 36. 

ing that it consists of two equal, opposite, and infinite branches, 
ADF and GEB, Fig. 36, to which a pair of straight lines, M 
and K, crossing at the centre C, are tangent only at an infinite 
distance from C. Such lines are called asymptotes. The fixed 
points R and K are called its foci, each one, & focus. 

The complete symbolism of this line is remarkable for its 
ready and striking truthfulness. 
6* 



130 FREE-HAND GEOMETRICAL DRAWING. 

The general idea of the infinite approach of a curve to a 
straight tangent, as a symbol of an infinite progress towards 
perfection, or to the absolute ideal, never actually attained, has 
long been familiar ; but is realized in the case of any of the 
many different curves which have asymptotes. The distinc- 
tive symbolism of the hyperbola may be more precisely stated 

Fii'st, there is material civilization, as that of peoples wh< 
excel more in material arts, than in personal or national virtues, 
and there is moral civilization, as of peoples or communities 
eminent for truth, justice, pure patriotism, and philanthropy. 
Also there is material barbarism ; and there is moral barbar- 
ism, illustrated by the injustice or cruelties practised upon 
weaker, or savage, peoples by nations who were far advanced 
in many material arts. 

Now, in the hyperbola, one asymptote, as M, may represent 
material perfection, or material degradation ; the former, for 
example, to the right, and the latter, to the left of C. The 
other asymptote, N, may then represent to the right of C moral 
perfection, and to the left, its opposite. Thus the two branches 
of the curve, each tangent to both asymptotes, naturally repre- 
sent the opposite possibilities of indefinite progress towards good 
or evil, either material or moral. 

101. Spirals are, as compared with the circle, noble symbols 
of immortal life, with growth and progress, inasmuch as, unlike 
the circle, they do not return into themselves, but ever proceed 
in wider and wider circuits, expressive of the expansive progress 
of all noble lives. 

They may, therefore, well enter into the composition of the 
decorative parts, at least, or the seals, or heraldic devices of 
the buildings whose uses are representative of human progress. 
And they could hardly appear otherwise than in the ornamental 
details, because the visible representative of the inspiring idea 
should be, like the idea itself, over and above the working 
rooms which must be merely adapted to the work to be done in 
them. 

102. Imagine now a curve, such that the positions of all its 
points should be governed by one fixed point and one fixed line. 

Together with such a curve, imagine any organization, the 



3- 



GEOMETRIC SYMBOLISM. 



131 



various brandies, or departments of whose work, should be gov- 
erned by some one central idea, and some one executive body, 
representing, so to speak, a certain line of policy. 

Such a curve would be a symbol of such an organization ; and, 
if, in future times, attention were paid to symbolism between 
the inward idea, or purpose, and the outward material agencies 
through which the idea was put in operation, in all departments 
of activity, as it has already been in some, nothing would seem 
more natural than endeavors to realize this symbolism. 

103. Thus, for a long time it has been customary to build 
churches in the form of a cross ; to decorate heroes with jewelled, 
and hence brilliant stars ; to mark a court-house (temple of 
justice) by a statue holding a balance ; to crown a building 
which is the property of a nation, or, in some sense, even of 
mankind, by a vast dome, expressive of the firmament under 
winch all live. 

101. With equal propriety, apparently nnthonght of only be- 
cause the field of application is much more recent, might sym- 
bolism enter the field of education. It does so on a small scale, 
when, for instance, a quill is made the device for the vane of 
a school-house, or an engineering instrument the device used 
for a breast-pin by the students of a school of engineering, or 
when the iron fence-posts around a military academy are in the 
form of cannon, and the pickets in the form of spears. But, on 
a larger scale, the buildings for the general and special pur- 
poses of an} r large educational establishment, together with the 
residences for its teaching body, might, 
if its grounds were sufficiently exten- 
sive, be easily arranged in a symbolical 
manner. 

105. Thus, returning now to Art. 
(102) there are two curves, at least, 
which agree with the definition there 
suggested. These are the parabola, Fig. 
37, each of whose points, as a, is at 
equal distances, a F and a b, from a 
fixed point, F, the focus, from a fixed 
line, D b, the directrix. 

Also the conchoid, Fig. 38, a curve of two branches, and all 
of whose points are at the same fixed distance from a given line, 




F&97. 



132 



FREE-HAND GEOMETRICAL DRAWING. 



measured on lines drawn from & fixed point. When this point 
is nearer the fixed line than the fixed distance, one branch of 




the curve will be looped. Thus E E is the fixed line, and A 
the fixed point. Then dC = de / ba = be, etc. 



GEOMETRIC SYMBOLISM. 133 

106. Turning now from these curves to a complex educa- 
tional establishment, we find for its corresponding fixed ele- 
ments, 1°, a foundation course of study, alike for all, 2°, a 
teaching and governing body. There would then be depend- 
ing upon these (a) the various specialties to which the institu- 
tion might devote itself; (b) the incidental features of its life, 
lodgings, gymnasium, etc. ; and finally (c) a select group of struc- 
tures, devoted to the most refined purposes of the institution. 

107. Parabolic plan. The several elements just stated 
might find their material organization on a parabolic plan, as 
follows. The collegiate or general building in which the foun- 
dation course (1°) should be given, would naturally stand at the 
focus. The residences of the teaching and governing body (2°) 
would be located on an avenue marking the directrix. Then, 
(a) the schools for the several specialties or professions, would 
be arranged at intervals on the curve, and with paths to them 
located as at F a and b a. The axis of symmetry, D F, being 
therefore a special line, and D and V, special points upon it, a 
chapel, D, library, V, a museum, and observatory may be built 
upon it. 

Subordinate structures might be located at convenience on 
lines within the curve, and parallel to D F. 

108. The conchoidal plan. The conchoid, when laid out on 
a grand scale on the ground, permits the symbolical expression 
of the ideas, stated in Art. (106), in the material organization of 
an institution, as its published curriculum exhibits them in the 
printed expression of the logical organization. 

1°. A grand building, surmounted with a dome, as the sym- 
bol of comprehensiveness, and with lofty porticos facing the 
four cardinal points of the compass, as the symbol of its equal 
openness to all, should stand at A, and contain instruction 
rooms for all the general subjects. 

2°. Professors, as the immediate personal determining ele- 
ment in the life and work of an institution, should have resi- 
dences ranged along the fixed determining line E E. And d e 
may be 1000 feet or more. 

3°. D D, being the superior branch of the curve, should be 
allotted to the series of buildings devoted to the several profes- 
sional schools, and reached from A by paths on the radial lines 
as a b c, which determine the points where they stand. 



134 FREE-HAND GEOMETRIC 1L DRAWING. 

4°. B B, being the inferior branch of the curve, should be 
devoted to the gymnasium, janitor's lodge, bathing house, lodg- 
ing, and society buildings. 

5°. The loop A e, as a separate and peculiar feature, should 
be set apart for an elegant enclosure, with fountains, etc., and 
faced by the observatory, chapel, and library buildings. 

109. Education, most comprehensively denned as to its mat- 
ter, exists in two grand divisions ; humanistic, or the study of 
man, his life, and actions ; and naturalistic, or the study of 
nature as subservient to man. 

If, according to what may be the preference of some, an in- 
stitution of the most comprehensive or encyclopedic character, 
embracing both of these grand divisions, were to be planned, 
its symbolic material organization might best consist in the ar- 
rangement of its buildings on the two branches of a hyperbola. 

But it is probably better, that, by a proper application in 
education of the principle of division of labor, only one of these 
grand divisions of the whole field should be embraced in one 
institution. If so, there would seem to be no superior to the 
conchoid for the symbolical ground plan of the buildings col- 
lectively, of a great educational establishment of either class. 

110. But the bi-lateral symmetry of the conchoid, that is, the 
equality of the halves on each side of the line of symmetry Ce, 
may be made significant in either of two ways. 

First. In case of the adoption of the all-inclusive organiza- 
tion above described, the buildings pertaining to the two grand 
divisions named, might be arranged ; those of the one, on one 
side, and those of the other, on the other side of Ce. 

Second, independently of this, and probably a better symbolic 
use of this symmetry, would be the following. Each subject of 
study has its purely scientific side, as related to truth ; and its 
artistic side, as related to beauty. Hence, taking, as before, the 
buildings of but one of the grand divisions described, for distri- 
bution on the conchoid plan, the naturalistic one for example ; 
Schools of Industrial Physics, Chemistry, and Engineering 
could be on one side of Ce, and those of Music, Painting, 
Architecture, and Decorative Design, on the other. 

111. The positions of lines have a significance, as well as 
their forms. Thus a prevalence of vertical lines symbolizes as- 



GEOMETRIC SYMBOLISM. 135 

piration, upward-tending thought and purpose ; and hence gives 
noble meaning to a lofty gothic cathedral interior, where the 
prevailing direction of the lines is vertical. 

The same idea gives effect to the humblest village spire. 
Hence the betrayal of offensive vain consciousness, or of obtuse- 
ness, either in the maker or beholder, in adding an up-pointing 
hand to the top of a spire, as if the spire were made to say, " See 
with what beautiful expressiveness I point to heaven ; " or, 
more likely, as if the mind could not understand the upward 
pointing of the spire without this explanatory addition, which 
robs the imagination of its dues in being left free to give mean- 
ing to what it sees. 

A prevalence of horizontal lines, is expressive of a clinging 
to the earth, as in the popular life of the Greeks, most, or all 
of whose gods were but exaggerated men, crimes and all ; and 
then, set over this world's woods and fields, seas and skies, wars 
and passions, rather than over a universe of life, to be moulded 
into enduring forms of living beauty by them. Hence the 
marked predominance of the horizontal in the Greek temples, 
with their flat roofs and horizontal mouldings, and flat doors and 
window tops. 

Once more, and in a derivative manner, horizontal lines ex- 
press firmness, decision, stability, and hence are the proper 
characteristic lines of foundations and supports. The repose, 



a: 



Fig. 39. Fig. 40. 

or fixity, which they primarily signify, leads to the secondary 
meanings, unchangeableness, and thence decision, or stability, 
as stated. Hence the curved outlines of mouldings on support- 
ing parts best flow into the horizontal top and bottom surfaces 
of such parts. 

Thus, Fiu;s. 39 and 41, show a better relation of the curved 
contour, as tangent to the bases, than Fig. 40, does. 



136 FREE-HAND GEOMETEICAL DRAWING. 

112. Carvings. Work becomes so costly as soon as straight 
outlines are abandoned, and especially as carved work begins 
to be employed, that its consequent difficulty of attainment 
makes it symbolical of the grace and beauty that can only be 
had under the best conditions, or, as the result of man's best 
aspirations; while the plain lines of ordinary work represent, 



Fig. 41. 

by comparison, humbler human industries. Hence a bit of 
choice carving to crown, or tip, or face a piece of otherwise 
plain work, happily symbolizes the cheerful co-operation of hap- 
piness and honest industry, the meeting of truth and beauty. 

It is in the light of such reflections that the real vulgarity of 
mere flat sawed scroll work, on which no elevated intellectual 
or artistic thought or fond purpose has been exercised, is fully 
shown. Being purely mechanical products, they can serve no 
high thought or purpose. 

113. An entirely (liferent principle, however, governs the 
employment of ornamental castings from really rich and beau- 
tiful designs. Here, the thought is the nobly generous one of 
bringing to every humble home, by means of a beneficent art 
of multiplication, beauties of decoration which could not other- 
wise be had. The " preciousness " of the immediate products 
of the skilled and refined hand becomes only their hatefulness 
when they are prized mainly because none but one wealthy pur- 
chaser can own and enjoy them. 

The " ginger-bread " products of the scroll saw, from inch 
boards, are mean in origin, material, and execution, and are 
therefore to be discarded for their inherent demerits ; but good 
castings, from beautiful designs, inherit and partake of the 



GEOMETRIC SYMBOLISM. 137 

characteristics and associations of their original, and are, by all 
means, to be commended, where originals cannot be had. 

Somewhat in the same line of thought with the remarks on 
carvings; broken pediments, as in the annexed figure, and con- 



^Lfci^ 



taining a carved bust or other form of life, may be mentioned 
as symbolizing the escape of the spirit from the hindrances and 
imprisoment of the body. 

114. Without further illustration, it may now be enough to 
add that the foregoing somewhat numerous, and w T idely varied 
examples may serve to set the thoughts in motion upon the line 
indicated, so that the student may be aided in his efforts to give 
to all his works an attractive and elevating meaning^ at the 
same time that they fulfil the bare physical conditions required 
of them. 

Apply the principles of this, and the preceding chapters, in 
designing: the following: : 



Ex. 


81. 


An altar. 


Ex. 


32. 


A pulpit. 


Ex. 


33. 


A book-case. 


Ex. 


34. 


A parlor organ case. 


Ex. 


36. 


A church porch. 


Ex. 


37. 


A district school-house. 


Ex. 


38. 


A sideboard. 


Ex. 


39. 


A public library entrance. 


Ex. 


40. 


A mantel-piece. 



THE END. 



